2577:
2143:
2572:{\displaystyle {\begin{aligned}\left&=\lbrace x\mid x\in \mathbb {R} ,a\leq x\leq b\rbrace \\\left&=\lbrace x\mid x\in \mathbb {R} ,a\leq x\rbrace \cup \lbrace \infty \rbrace \\\left&=\lbrace x\mid x\in \mathbb {R} ,b\leq x\rbrace \cup \lbrace \infty \rbrace \cup \lbrace x\mid x\in \mathbb {R} ,x\leq a\rbrace \\\left&=\lbrace \infty \rbrace \cup \lbrace x\mid x\in \mathbb {R} ,x\leq a\rbrace \\\left&=\{a\}\\\left&=\lbrace \infty \rbrace \end{aligned}}}
1255:
148:
937:
51:
1990:
1250:{\displaystyle {\begin{aligned}a+\infty =\infty +a&=\infty ,&a\neq \infty \\a-\infty =\infty -a&=\infty ,&a\neq \infty \\a/\infty =a\cdot 0=0\cdot a&=0,&a\neq \infty \\\infty /a&=\infty ,&a\neq \infty \\a/0=a\cdot \infty =\infty \cdot a&=\infty ,&a\neq 0\\0/a&=0,&a\neq 0\end{aligned}}}
1721:
5291:
1376:
1783:
1265:
The following expressions cannot be motivated by considering limits of real functions, and no definition of them allows the statement of the standard algebraic properties to be retained unchanged in form for all defined cases. Consequently, they are left undefined:
2840:
1511:
5588:
5173:
6190:
4003:
1272:
3007:. The result of an arithmetic operation on intervals is always an interval, except when the intervals with a binary operation contain incompatible values leading to an undefined result. In particular, we have, for every
3615:
1985:{\displaystyle {\begin{aligned}a\cdot (b+c)&=a\cdot b+a\cdot c\\a&=\left({\frac {a}{b}}\right)\cdot b&=\,\,&{\frac {(a\cdot b)}{b}}\\a&=(a+b)-b&=\,\,&(a-b)+b\end{aligned}}}
7106:
5050:
4981:
4920:
4851:
5724:
6609:
5095:
3280:
3144:
2148:
1788:
1776:
1516:
1504:
1277:
942:
6980:
5770:
4614:
4497:
3048:
4790:
3654:
3691:
3243:
929:
4670:
4553:
4724:
249:
6645:
6514:
6937:
6334:
6272:
6032:
5681:
4261:
4201:
2135:
312:
7031:
6810:
6479:
5998:
5967:
3198:
2983:
2694:
2652:
2617:
2089:
2046:
1716:{\displaystyle {\begin{aligned}(a+b)+c&=a+(b+c)\\a+b&=b+a\\(a\cdot b)\cdot c&=a\cdot (b\cdot c)\\a\cdot b&=b\cdot a\\a\cdot \infty &={\frac {a}{0}}\\\end{aligned}}}
1440:
892:
670:
617:
561:
6550:
5497:
5454:
4436:
4383:
4141:
3776:
2725:
2582:
With the exception of when the end-points are equal, the corresponding open and half-open intervals are defined by removing the respective endpoints. This redefinition is useful in
419:
6880:
5512:
4327:
2874:
811:
278:
6419:
6371:
6238:
2942:
7000:
6572:
6300:
5371:
5286:{\displaystyle f:{\widehat {\mathbb {R} }}\to {\widehat {\mathbb {R} }},A\subseteq {\widehat {\mathbb {R} }},L\in {\widehat {\mathbb {R} }},p\in {\widehat {\mathbb {R} }}}
4287:
3508:
3005:
2720:
2015:
861:
639:
211:
3540:
5409:
4086:
3856:
6442:
6394:
6118:
6079:
5166:
68:
6056:
3429:
3353:
2903:
5805:
5137:
4048:
3821:
7138:
7063:
6213:
6110:
5896:
3465:
3389:
1409:
5846:
2091:. However, since it is not an ordered set, the interval has a slightly different meaning. The definitions for closed intervals are as follows (it is assumed that
1371:{\displaystyle {\begin{aligned}&\infty +\infty \\&\infty -\infty \\&\infty \cdot 0\\&0\cdot \infty \\&\infty /\infty \\&0/0\end{aligned}}}
7199:
3891:
7293:
7240:
3563:
115:
7303:
719:(also called linear fractional transformations), with the understanding that when the denominator of the linear fractional transformation is
87:
6482:
94:
826:
The arithmetic operations on this space are an extension of the same operations on reals. A motivation for the new definitions is the
505:
362:
134:
2919:
on this circle (either measured straight or along the circle). There is no metric which is an extension of the ordinary metric on
4292:
The definitions given above can be compared with the usual definitions of limits of real functions. In the following statements,
101:
7068:
509:
72:
4986:
4926:
4856:
4796:
83:
5689:
6681:
relation between points is part of the structure of the real projective line. For instance, given any pair of points, the
6678:
6577:
5063:
3248:
3056:
465:
is undefined, even though the reciprocal is total. It has usable interpretations, however – for example, in geometry, the
1729:
1457:
7335:
7330:
6942:
6709:
5732:
4559:
4442:
3290:
3010:
4729:
3619:
3659:
3211:
897:
4619:
4502:
176:
4676:
220:
6732:
6618:
6487:
5457:
39:
6885:
6781:
An extension does however exist in which all the algebraic properties, when restricted to defined operations in
6305:
6243:
6003:
5621:
4206:
4146:
2094:
283:
7005:
6784:
6757:
6724:
6453:
5972:
5941:
3172:
2957:
2835:{\displaystyle (b,a)=\{x\mid x\in \mathbb {R} ,b<x\}\cup \{\infty \}\cup \{x\mid x\in \mathbb {R} ,x<a\}}
2668:
2626:
2591:
2063:
2020:
1414:
866:
778:
644:
591:
535:
156:
6519:
5583:{\displaystyle f:{\widehat {\mathbb {R} }}\to {\widehat {\mathbb {R} }},\quad p\in {\widehat {\mathbb {R} }}.}
5467:
5424:
716:
108:
6341:
4388:
4335:
4093:
3728:
3474:
391:
61:
6828:
7325:
6740:
4295:
2057:
440:
437:
2845:
6752:
6670:
5414:
6674:
787:
254:
6666:
6612:
6337:
2916:
1382:
704:
697:
346:
30:
This article is about the extension of the reals by a single point at infinity. For the extension by
6399:
6351:
6277:
6218:
6035:
5594:
3697:
2951:
2922:
2583:
827:
759:
482:
454:
323:
6985:
6555:
6283:
5344:
4270:
3484:
2988:
2703:
1998:
844:
622:
194:
6705:
6185:{\displaystyle \tan \left({\frac {\pi }{2}}+n\pi \right)=\infty {\text{ for }}n\in \mathbb {Z} ,}
3513:
588:
can't be compared with any of the other elements, there's no point in retaining this relation on
5376:
4053:
3826:
151:
The projectively extended real line can be visualized as the real number line wrapped around a
7340:
7299:
6682:
6424:
6376:
6082:
6061:
5901:
5418:
5142:
2662:
763:
708:
701:
689:
315:
184:
160:
6041:
3408:
3332:
2879:
251:
with the standard arithmetic operations extended where possible, and is sometimes denoted by
6736:
6735:. Two of these correspond to elementary, arithmetic operations on the real projective line:
6660:
6089:
5784:
5116:
4027:
3800:
2697:
458:
382:
339:
7111:
7036:
6396:
As seen above, the tangent function can be prolongated to a function that is continuous in
6198:
6095:
5869:
3438:
3362:
1387:
6743:. Indeed, 0 and ∞ are fixed under negation, while 1 and −1 are fixed under reciprocation.
5822:
3862:
2658:
781:
774:
3998:{\displaystyle \lim _{x\to p^{+}}{f(x)}=L\qquad \left(\lim _{x\to p^{-}}{f(x)}=L\right),}
766:
on the real projective line. This cannot be extended to 4-tuples of points, because the
7152:
494:
490:
450:
331:
319:
7319:
168:
147:
6813:
6762:
6697:
6693:
2908:
712:
693:
529:
4329:
the first limit is as defined above, and the second limit is in the usual sense:
5102:
767:
486:
188:
50:
6700:
of the real projective line. The projectivities are described algebraically as
17:
6701:
6648:
5935:
2912:
730:
The detailed analysis of the action shows that for any three distinct points
2620:
180:
3542:
is a neighbourhood (resp. a right-sided or a left-sided neighbourhood) of
6713:
6686:
6447:
3610:{\displaystyle f:{\widehat {\mathbb {R} }}\to {\widehat {\mathbb {R} }},}
3166:
1726:
The following is true whenever expressions involved are defined, for any
777:
is appropriate, because the points are in 1-to-1 correspondence with one-
335:
327:
6982:(where the interval on either side is defined), all intervals excluding
6240:
but cannot be prolongated further to a function that is continuous in
7209:
is undefined, the result of division of these intervals is undefined.
1452:
Either both sides are undefined, or both sides are defined and equal.
839:
152:
466:
381:
Unlike most mathematical models of numbers, this structure allows
361:. The projectively extended real number line is distinct from the
146:
7268:
3477:(resp. a right-sided or a left-sided punctured neighbourhood) of
7245:. Directorate of Distance Education, University of North Bengal.
6345:
3200:. The definitions are motivated by the topology of this space.
322:
of the real line. More precisely, the point at infinity is the
4016:, there is a right-sided (left-sided) punctured neighbourhood
44:
6446:
Many discontinuous functions that become continuous when the
349:
in which three points have been assigned the specific values
345:
The projectively extended real line may be identified with a
6723:
The projectivities which are their own inverses are called
6302:
cannot be prolongated to functions that are continuous in
5938:, can be prolongated, in a unique way, to a function from
4266:
2915:
corresponding (for a given homeomorphism) to the ordinary
6825:
If consistency of complementation is required, such that
6552:
On the other hand, some functions that are continuous in
7101:{\displaystyle {\widehat {\mathbb {R} }}\setminus \{a\}}
6481:
remain discontinuous if the codomain is extended to the
7033:
may be naturally represented using this notation, with
5045:{\displaystyle \lim _{x\to +\infty }{|f(x)|}=+\infty }
4976:{\displaystyle \lim _{x\to \infty ^{-}}{f(x)}=\infty }
4915:{\displaystyle \lim _{x\to -\infty }{|f(x)|}=+\infty }
4846:{\displaystyle \lim _{x\to \infty ^{+}}{f(x)}=\infty }
1995:
In general, all laws of arithmetic that are valid for
7155:
7114:
7108:, and half-open intervals with equal endpoints, e.g.
7071:
7039:
7008:
6988:
6945:
6888:
6831:
6787:
6677:
are implicit. In particular, the construction of the
6621:
6580:
6558:
6522:
6490:
6456:
6427:
6402:
6379:
6354:
6308:
6286:
6246:
6221:
6201:
6121:
6098:
6064:
6044:
6006:
5975:
5944:
5872:
5825:
5787:
5735:
5719:{\displaystyle A\subseteq {\widehat {\mathbb {R} }},}
5692:
5624:
5515:
5470:
5427:
5379:
5347:
5176:
5145:
5119:
5066:
4989:
4929:
4859:
4799:
4732:
4679:
4622:
4562:
4505:
4445:
4391:
4338:
4298:
4273:
4209:
4149:
4096:
4056:
4030:
3894:
3829:
3803:
3731:
3662:
3622:
3566:
3516:
3487:
3441:
3411:
3365:
3335:
3251:
3214:
3175:
3059:
3013:
2991:
2960:
2925:
2882:
2848:
2728:
2706:
2671:
2629:
2594:
2146:
2097:
2066:
2023:
2001:
1786:
1732:
1514:
1460:
1417:
1390:
1275:
940:
900:
869:
847:
790:
742:, there is a linear fractional transformation taking
647:
625:
594:
538:
394:
286:
257:
223:
197:
6604:{\displaystyle \infty \in {\widehat {\mathbb {R} }}}
5090:{\displaystyle A\subseteq {\widehat {\mathbb {R} }}}
3275:{\displaystyle A\subseteq {\widehat {\mathbb {R} }}}
3139:{\displaystyle x\in \iff {\frac {1}{x}}\in \left\!,}
2048:
whenever all the occurring expressions are defined.
453:
in this structure. The structure, however, is not a
1771:{\displaystyle a,b,c\in {\widehat {\mathbb {R} }}.}
1499:{\displaystyle a,b,c\in {\widehat {\mathbb {R} }}.}
570:, there is no convincing argument to define either
75:. Unsourced material may be challenged and removed.
7193:
7132:
7100:
7057:
7025:
6994:
6974:
6931:
6874:
6804:
6639:
6603:
6566:
6544:
6508:
6473:
6436:
6413:
6388:
6365:
6328:
6294:
6266:
6232:
6207:
6184:
6104:
6073:
6050:
6026:
5992:
5961:
5890:
5840:
5799:
5764:
5718:
5675:
5582:
5491:
5448:
5403:
5365:
5285:
5160:
5131:
5089:
5044:
4975:
4914:
4845:
4784:
4718:
4664:
4608:
4547:
4491:
4430:
4377:
4321:
4281:
4255:
4195:
4135:
4080:
4042:
3997:
3850:
3815:
3770:
3685:
3648:
3609:
3534:
3502:
3459:
3423:
3383:
3347:
3274:
3237:
3192:
3138:
3042:
2999:
2977:
2936:
2897:
2868:
2834:
2714:
2688:
2646:
2611:
2571:
2129:
2083:
2040:
2009:
1984:
1770:
1715:
1498:
1434:
1403:
1370:
1249:
923:
886:
855:
805:
664:
633:
611:
555:
413:
318:, because it is considered as a neighbour of both
306:
272:
243:
205:
3149:irrespective of whether either interval includes
3132:
497:, by adding a single point called conventionally
6975:{\displaystyle a,b\in {\widehat {\mathbb {R} }}}
5765:{\displaystyle f:A\to {\widehat {\mathbb {R} }}}
5626:
4991:
4931:
4861:
4801:
4734:
4681:
4624:
4609:{\displaystyle \lim _{x\to \infty ^{-}}{f(x)}=L}
4564:
4507:
4492:{\displaystyle \lim _{x\to \infty ^{+}}{f(x)}=L}
4447:
4393:
4340:
4211:
4151:
4098:
3945:
3896:
3733:
3043:{\displaystyle a,b\in {\widehat {\mathbb {R} }}}
481:The projectively extended real line extends the
7295:Geometry: from Isometries to Special Relativity
6421:but this function cannot be made continuous at
4785:{\displaystyle \lim _{x\to p}{|f(x)|}=+\infty }
3649:{\displaystyle p\in {\widehat {\mathbb {R} }},}
461:arithmetic operations are total – for example,
6696:preserve the harmonic relation, they form the
6685:is the projective harmonic conjugate of their
3686:{\displaystyle L\in {\widehat {\mathbb {R} }}}
3238:{\displaystyle x\in {\widehat {\mathbb {R} }}}
924:{\displaystyle a\in {\widehat {\mathbb {R} }}}
838:In addition to the standard operations on the
6708:, according to the general construction of a
4665:{\displaystyle \lim _{x\to +\infty }{f(x)}=L}
4548:{\displaystyle \lim _{x\to -\infty }{f(x)}=L}
1261:Arithmetic operations that are left undefined
8:
7095:
7089:
5483:
5477:
5440:
5434:
4719:{\displaystyle \lim _{x\to p}{f(x)}=\infty }
3529:
3523:
2829:
2797:
2791:
2785:
2779:
2747:
2562:
2556:
2523:
2517:
2484:
2452:
2446:
2440:
2407:
2375:
2369:
2363:
2357:
2325:
2292:
2286:
2280:
2248:
2215:
2177:
244:{\displaystyle \mathbb {R} \cup \{\infty \}}
238:
232:
27:Real numbers with an added point at infinity
6640:{\displaystyle {\overline {\mathbb {R} }}.}
6509:{\displaystyle {\overline {\mathbb {R} }}.}
2586:when dividing by an interval containing 0.
894:, the following operations are defined for
6932:{\displaystyle (a,b]^{\complement }=(b,a]}
6329:{\displaystyle {\widehat {\mathbb {R} }}.}
6267:{\displaystyle {\widehat {\mathbb {R} }}.}
6027:{\displaystyle {\widehat {\mathbb {R} }}.}
5676:{\displaystyle \lim _{x\to p}{f(x)}=f(p).}
4256:{\displaystyle \lim _{x\to p^{-}}{f(x)}=L}
4196:{\displaystyle \lim _{x\to p^{+}}{f(x)}=L}
3085:
3081:
2130:{\displaystyle a,b\in \mathbb {R} ,a<b}
307:{\displaystyle {\widehat {\mathbb {R} }}.}
7171:
7154:
7113:
7076:
7075:
7073:
7072:
7070:
7038:
7026:{\displaystyle {\widehat {\mathbb {R} }}}
7013:
7012:
7010:
7009:
7007:
6987:
6962:
6961:
6959:
6958:
6944:
6905:
6887:
6848:
6830:
6805:{\displaystyle {\widehat {\mathbb {R} }}}
6792:
6791:
6789:
6788:
6786:
6625:
6624:
6622:
6620:
6591:
6590:
6588:
6587:
6579:
6560:
6559:
6557:
6529:
6521:
6494:
6493:
6491:
6489:
6474:{\displaystyle {\widehat {\mathbb {R} }}}
6461:
6460:
6458:
6457:
6455:
6426:
6404:
6403:
6401:
6378:
6356:
6355:
6353:
6313:
6312:
6310:
6309:
6307:
6288:
6287:
6285:
6251:
6250:
6248:
6247:
6245:
6223:
6222:
6220:
6200:
6175:
6174:
6163:
6133:
6120:
6097:
6063:
6043:
6011:
6010:
6008:
6007:
6005:
5993:{\displaystyle {\widehat {\mathbb {R} }}}
5980:
5979:
5977:
5976:
5974:
5962:{\displaystyle {\widehat {\mathbb {R} }}}
5949:
5948:
5946:
5945:
5943:
5871:
5824:
5786:
5752:
5751:
5749:
5748:
5734:
5703:
5702:
5700:
5699:
5691:
5641:
5629:
5623:
5567:
5566:
5564:
5563:
5543:
5542:
5540:
5539:
5526:
5525:
5523:
5522:
5514:
5469:
5426:
5378:
5346:
5325:, if and only if for every neighbourhood
5273:
5272:
5270:
5269:
5250:
5249:
5247:
5246:
5227:
5226:
5224:
5223:
5204:
5203:
5201:
5200:
5187:
5186:
5184:
5183:
5175:
5144:
5118:
5077:
5076:
5074:
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5065:
5027:
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5009:
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4988:
4953:
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4928:
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4880:
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4815:
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4749:
4737:
4731:
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4642:
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4621:
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4567:
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4525:
4510:
4504:
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4461:
4450:
4444:
4408:
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4390:
4355:
4343:
4337:
4312:
4311:
4297:
4275:
4274:
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4233:
4225:
4214:
4208:
4173:
4165:
4154:
4148:
4113:
4101:
4095:
4055:
4029:
3967:
3959:
3948:
3918:
3910:
3899:
3893:
3828:
3802:
3748:
3736:
3730:
3673:
3672:
3670:
3669:
3661:
3633:
3632:
3630:
3629:
3621:
3594:
3593:
3591:
3590:
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3576:
3574:
3573:
3565:
3515:
3486:
3440:
3410:
3364:
3334:
3262:
3261:
3259:
3258:
3250:
3225:
3224:
3222:
3221:
3213:
3193:{\displaystyle {\widehat {\mathbb {R} }}}
3180:
3179:
3177:
3176:
3174:
3117:
3104:
3086:
3058:
3030:
3029:
3027:
3026:
3012:
2993:
2992:
2990:
2978:{\displaystyle {\widehat {\mathbb {R} }}}
2965:
2964:
2962:
2961:
2959:
2927:
2926:
2924:
2881:
2862:
2861:
2847:
2813:
2812:
2763:
2762:
2727:
2708:
2707:
2705:
2689:{\displaystyle {\widehat {\mathbb {R} }}}
2676:
2675:
2673:
2672:
2670:
2647:{\displaystyle {\widehat {\mathbb {R} }}}
2634:
2633:
2631:
2630:
2628:
2612:{\displaystyle {\widehat {\mathbb {R} }}}
2599:
2598:
2596:
2595:
2593:
2468:
2467:
2391:
2390:
2341:
2340:
2264:
2263:
2193:
2192:
2147:
2145:
2111:
2110:
2096:
2084:{\displaystyle {\widehat {\mathbb {R} }}}
2071:
2070:
2068:
2067:
2065:
2041:{\displaystyle {\widehat {\mathbb {R} }}}
2028:
2027:
2025:
2024:
2022:
2003:
2002:
2000:
1952:
1951:
1887:
1884:
1883:
1858:
1787:
1785:
1755:
1754:
1752:
1751:
1731:
1699:
1515:
1513:
1483:
1482:
1480:
1479:
1459:
1435:{\displaystyle {\widehat {\mathbb {R} }}}
1422:
1421:
1419:
1418:
1416:
1395:
1389:
1356:
1340:
1276:
1274:
1211:
1148:
1109:
1046:
941:
939:
911:
910:
908:
907:
899:
887:{\displaystyle {\widehat {\mathbb {R} }}}
874:
873:
871:
870:
868:
849:
848:
846:
797:
793:
792:
789:
688:is the way the real projective line is a
665:{\displaystyle {\widehat {\mathbb {R} }}}
652:
651:
649:
648:
646:
627:
626:
624:
612:{\displaystyle {\widehat {\mathbb {R} }}}
599:
598:
596:
595:
593:
556:{\displaystyle {\widehat {\mathbb {R} }}}
543:
542:
540:
539:
537:
395:
393:
291:
290:
288:
287:
285:
264:
260:
259:
256:
225:
224:
222:
199:
198:
196:
135:Learn how and when to remove this message
6545:{\displaystyle x\mapsto {\frac {1}{x}}.}
5492:{\displaystyle A\cup \lbrace p\rbrace .}
5449:{\displaystyle A\cup \lbrace p\rbrace ,}
762:of linear fractional transformations is
512:of the real line) distinguishes between
7222:
7086:
6989:
6774:
4008:if and only if for every neighbourhood
3781:if and only if for every neighbourhood
6336:This is the case, for example, of the
5109:if and only if every neighbourhood of
4431:{\displaystyle \lim _{x\to p}{f(x)}=L}
4378:{\displaystyle \lim _{x\to p}{f(x)}=L}
4136:{\displaystyle \lim _{x\to p}{f(x)}=L}
3771:{\displaystyle \lim _{x\to p}{f(x)}=L}
834:Arithmetic operations that are defined
414:{\displaystyle {\frac {a}{0}}=\infty }
6875:{\displaystyle ^{\complement }=(b,a)}
6812:, resolve to the standard rules: see
5333:, there is a punctured neighbourhood
3789:, there is a punctured neighbourhood
7:
7269:"Projectively Extended Real Numbers"
7262:
7260:
7258:
7256:
7254:
7252:
7234:
7232:
7230:
7228:
7226:
7149:For example, the ratio of intervals
6669:is considered in the context of the
6483:affinely extended real number system
6373:but it cannot be made continuous at
5415:topological definition of continuity
4322:{\displaystyle p,L\in \mathbb {R} ,}
3169:can be used to analyze functions of
822:Motivation for arithmetic operations
563:in a meaningful way. Given a number
73:adding citations to reliable sources
6712:. Collectively they form the group
6034:In particular, this is the case of
2869:{\displaystyle a,b\in \mathbb {R} }
6581:
6428:
6380:
6160:
6065:
6045:
5039:
5004:
4970:
4942:
4909:
4874:
4840:
4812:
4779:
4713:
4637:
4575:
4520:
4458:
3321:is a right-sided neighbourhood of
2788:
2559:
2541:
2535:
2443:
2419:
2366:
2289:
2233:
1689:
1345:
1337:
1329:
1309:
1301:
1295:
1287:
1281:
1187:
1171:
1165:
1138:
1124:
1106:
1099:
1051:
1036:
1022:
1006:
1000:
987:
973:
957:
951:
506:affinely extended real number line
408:
363:affinely extended real number line
235:
25:
6516:This is the case of the function
3397:is a left-sided neighbourhood of
84:"Projectively extended real line"
6704:, since the real numbers form a
5413:This corresponds to the regular
3435:contains the semi-open interval
3359:contains the semi-open interval
2696:. Sufficient for a base are the
931:, with exceptions as indicated:
806:{\displaystyle \mathbb {R} ^{2}}
273:{\displaystyle \mathbb {R} ^{*}}
49:
5556:
3938:
1450:The following equalities mean:
532:relation cannot be extended to
173:projectively extended real line
60:needs additional citations for
7188:
7176:
7168:
7156:
7127:
7115:
7052:
7040:
6926:
6914:
6902:
6889:
6869:
6857:
6845:
6832:
6526:
5882:
5876:
5835:
5829:
5745:
5667:
5661:
5651:
5645:
5633:
5536:
5389:
5383:
5197:
5028:
5024:
5018:
5011:
4998:
4963:
4957:
4938:
4898:
4894:
4888:
4881:
4868:
4833:
4827:
4808:
4768:
4764:
4758:
4751:
4741:
4706:
4700:
4688:
4652:
4646:
4631:
4596:
4590:
4571:
4535:
4529:
4514:
4479:
4473:
4454:
4418:
4412:
4400:
4365:
4359:
4347:
4243:
4237:
4218:
4183:
4177:
4158:
4123:
4117:
4105:
4066:
4060:
3977:
3971:
3952:
3928:
3922:
3903:
3839:
3833:
3758:
3752:
3740:
3587:
3454:
3442:
3378:
3366:
3082:
3078:
3066:
2741:
2729:
1969:
1957:
1937:
1925:
1902:
1890:
1809:
1797:
1647:
1635:
1613:
1601:
1565:
1553:
1531:
1519:
830:of functions of real numbers.
314:The added point is called the
1:
7205:in both intervals, and since
6679:projective harmonic conjugate
6414:{\displaystyle \mathbb {R} ,}
6366:{\displaystyle \mathbb {R} ,}
6233:{\displaystyle \mathbb {R} ,}
5056:Extended definition of limits
2937:{\displaystyle \mathbb {R} .}
696:to a circle. For example the
680:Fundamental to the idea that
7292:Lee, Nam-Hoon (2020-04-28).
6995:{\displaystyle \varnothing }
6629:
6567:{\displaystyle \mathbb {R} }
6498:
6295:{\displaystyle \mathbb {R} }
5366:{\displaystyle x\in A\cap C}
4282:{\displaystyle \mathbb {R} }
3503:{\displaystyle x\not \in A,}
3401:, if there is a real number
3325:, if there is a real number
3000:{\displaystyle \mathbb {R} }
2715:{\displaystyle \mathbb {R} }
2654:excluding any single point.
2010:{\displaystyle \mathbb {R} }
856:{\displaystyle \mathbb {R} }
634:{\displaystyle \mathbb {R} }
206:{\displaystyle \mathbb {R} }
6710:projective line over a ring
6673:, then the consequences of
3556:Basic definitions of limits
3535:{\displaystyle A\cup \{x\}}
686:no different from any other
508:(also called the two-point
477:Extensions of the real line
183:), is the extension of the
7357:
6658:
6348:function is continuous in
5781:if and only if, for every
5404:{\displaystyle f(x)\in B.}
4267:Comparison with limits in
4081:{\displaystyle f(x)\in A.}
3305:contains an open interval
2623:are also intervals, as is
700:of 2 × 2 real
641:is used in definitions in
177:one-point compactification
29:
6647:This is the case for the
6611:become continuous if the
3881:from the right (left) is
3851:{\displaystyle f(x)\in A}
2907:As said, the topology is
489:in the same way that the
40:Extended real number line
6758:Complex projective plane
6437:{\displaystyle \infty .}
6389:{\displaystyle \infty .}
6074:{\displaystyle \infty ,}
5161:{\displaystyle y\neq p.}
2911:to a circle. Thus it is
2657:The open intervals as a
157:stereographic projection
7239:NBU, DDE (2019-11-05).
6342:trigonometric functions
6280:that are continuous in
6051:{\displaystyle \infty }
6038:, which take the value
3475:punctured neighbourhood
3424:{\displaystyle y\neq x}
3348:{\displaystyle y\neq x}
2898:{\displaystyle a<b.}
7195:
7140:, remaining undefined.
7134:
7102:
7059:
7027:
6996:
6976:
6933:
6876:
6806:
6641:
6605:
6568:
6546:
6510:
6475:
6438:
6415:
6390:
6367:
6330:
6296:
6268:
6234:
6209:
6186:
6106:
6075:
6052:
6028:
6000:that is continuous in
5994:
5963:
5892:
5842:
5801:
5800:{\displaystyle p\in A}
5766:
5720:
5677:
5584:
5493:
5450:
5405:
5367:
5287:
5162:
5133:
5132:{\displaystyle y\in A}
5091:
5046:
4977:
4916:
4847:
4786:
4720:
4666:
4610:
4549:
4493:
4432:
4379:
4323:
4283:
4257:
4197:
4137:
4082:
4044:
4043:{\displaystyle x\in B}
3999:
3852:
3817:
3816:{\displaystyle x\in B}
3772:
3687:
3650:
3611:
3536:
3504:
3461:
3425:
3385:
3349:
3276:
3239:
3194:
3140:
3044:
3001:
2979:
2938:
2899:
2870:
2836:
2716:
2690:
2648:
2613:
2573:
2131:
2085:
2052:Intervals and topology
2042:
2011:
1986:
1772:
1717:
1500:
1436:
1411:cannot be extended to
1405:
1372:
1251:
925:
888:
857:
807:
717:Möbius transformations
666:
635:
613:
557:
469:of a vertical line is
415:
330:of real numbers whose
308:
274:
245:
207:
164:
7273:mathworld.wolfram.com
7196:
7135:
7133:{\displaystyle (a,a]}
7103:
7065:being interpreted as
7060:
7058:{\displaystyle (a,a)}
7028:
6997:
6977:
6934:
6877:
6807:
6753:Real projective plane
6729:hyperbolic involution
6671:real projective plane
6655:As a projective range
6642:
6606:
6574:and discontinuous at
6569:
6547:
6511:
6476:
6439:
6416:
6391:
6368:
6331:
6297:
6269:
6235:
6210:
6208:{\displaystyle \tan }
6187:
6107:
6105:{\displaystyle \tan }
6076:
6053:
6029:
5995:
5964:
5893:
5891:{\displaystyle f(p).}
5843:
5802:
5767:
5721:
5678:
5585:
5494:
5451:
5406:
5368:
5288:
5163:
5134:
5092:
5047:
4978:
4917:
4848:
4787:
4721:
4667:
4611:
4550:
4494:
4433:
4380:
4324:
4284:
4258:
4198:
4138:
4090:It can be shown that
4083:
4045:
4000:
3853:
3818:
3773:
3688:
3651:
3612:
3537:
3505:
3462:
3460:{\displaystyle (y,x]}
3426:
3386:
3384:{\displaystyle [x,y)}
3350:
3277:
3240:
3195:
3141:
3045:
3002:
2980:
2939:
2900:
2871:
2837:
2717:
2691:
2649:
2614:
2574:
2132:
2086:
2043:
2012:
1987:
1773:
1718:
1501:
1454:This is true for any
1437:
1406:
1404:{\displaystyle e^{x}}
1373:
1252:
926:
889:
858:
817:Arithmetic operations
808:
667:
636:
614:
558:
493:extends the field of
416:
309:
275:
246:
217:. It is thus the set
213:, by a point denoted
208:
159:) with an additional
150:
7153:
7112:
7069:
7037:
7006:
6986:
6943:
6886:
6829:
6785:
6667:real projective line
6619:
6578:
6556:
6520:
6488:
6454:
6425:
6400:
6377:
6352:
6338:exponential function
6306:
6284:
6278:elementary functions
6244:
6219:
6199:
6119:
6112:is extended so that
6096:
6062:
6042:
6036:polynomial functions
6004:
5973:
5942:
5870:
5841:{\displaystyle f(x)}
5823:
5785:
5733:
5690:
5622:
5513:
5468:
5425:
5377:
5345:
5174:
5143:
5117:
5064:
4987:
4927:
4857:
4797:
4730:
4677:
4620:
4560:
4503:
4443:
4389:
4336:
4296:
4271:
4207:
4147:
4143:if and only if both
4094:
4054:
4028:
3892:
3827:
3801:
3729:
3660:
3620:
3564:
3514:
3485:
3439:
3409:
3363:
3333:
3249:
3212:
3173:
3057:
3011:
2989:
2958:
2923:
2880:
2846:
2726:
2704:
2669:
2627:
2592:
2144:
2095:
2064:
2021:
1999:
1784:
1730:
1512:
1458:
1446:Algebraic properties
1415:
1388:
1383:exponential function
1273:
938:
898:
867:
845:
788:
715:may be expressed by
698:general linear group
645:
623:
619:. However, order on
592:
536:
392:
347:real projective line
284:
255:
221:
195:
69:improve this article
7336:Projective geometry
7298:. Springer Nature.
7267:Weisstein, Eric W.
6344:. For example, the
2952:Interval arithmetic
2947:Interval arithmetic
2584:interval arithmetic
2060:can be extended to
2017:are also valid for
7331:Topological spaces
7191:
7130:
7098:
7055:
7023:
6992:
6972:
6929:
6872:
6802:
6675:Desargues' theorem
6637:
6601:
6564:
6542:
6506:
6471:
6434:
6411:
6386:
6363:
6326:
6292:
6264:
6230:
6205:
6182:
6102:
6071:
6048:
6024:
5990:
5959:
5888:
5838:
5797:
5762:
5716:
5673:
5640:
5580:
5489:
5446:
5401:
5363:
5283:
5158:
5129:
5087:
5042:
5008:
4973:
4952:
4912:
4878:
4843:
4822:
4782:
4748:
4716:
4695:
4662:
4641:
4606:
4585:
4545:
4524:
4489:
4468:
4428:
4407:
4375:
4354:
4319:
4279:
4253:
4232:
4193:
4172:
4133:
4112:
4078:
4040:
3995:
3966:
3917:
3848:
3813:
3768:
3747:
3683:
3646:
3607:
3532:
3500:
3457:
3421:
3381:
3345:
3272:
3235:
3190:
3136:
3040:
2997:
2975:
2934:
2895:
2866:
2832:
2722:and the intervals
2712:
2700:open intervals in
2686:
2644:
2609:
2569:
2567:
2127:
2081:
2056:The concept of an
2038:
2007:
1982:
1980:
1768:
1713:
1711:
1496:
1432:
1401:
1368:
1366:
1247:
1245:
921:
884:
853:
803:
662:
631:
609:
553:
457:, and none of the
411:
304:
270:
241:
203:
165:
7305:978-3-030-42101-4
7194:{\displaystyle /}
7083:
7020:
6969:
6799:
6683:point at infinity
6632:
6598:
6537:
6501:
6468:
6320:
6258:
6215:is continuous in
6166:
6141:
6018:
5987:
5956:
5902:rational function
5819:and the limit of
5775:is continuous in
5759:
5710:
5625:
5574:
5550:
5533:
5419:subspace topology
5417:, applied to the
5297:a limit point of
5280:
5257:
5234:
5211:
5194:
5113:includes a point
5084:
4990:
4983:is equivalent to
4930:
4860:
4853:is equivalent to
4800:
4733:
4726:is equivalent to
4680:
4623:
4616:is equivalent to
4563:
4506:
4499:is equivalent to
4446:
4392:
4385:is equivalent to
4339:
4210:
4150:
4097:
3944:
3895:
3732:
3680:
3640:
3601:
3584:
3269:
3232:
3187:
3125:
3112:
3094:
3037:
2972:
2683:
2641:
2606:
2078:
2035:
1909:
1866:
1762:
1707:
1490:
1429:
918:
881:
764:triply transitive
709:transitive action
690:homogeneous space
659:
606:
550:
504:In contrast, the
428:. In particular,
403:
316:point at infinity
298:
175:(also called the
161:point at infinity
155:(by some form of
145:
144:
137:
119:
16:(Redirected from
7348:
7310:
7309:
7289:
7283:
7282:
7280:
7279:
7264:
7247:
7246:
7236:
7210:
7208:
7204:
7200:
7198:
7197:
7192:
7175:
7147:
7141:
7139:
7137:
7136:
7131:
7107:
7105:
7104:
7099:
7085:
7084:
7079:
7074:
7064:
7062:
7061:
7056:
7032:
7030:
7029:
7024:
7022:
7021:
7016:
7011:
7001:
6999:
6998:
6993:
6981:
6979:
6978:
6973:
6971:
6970:
6965:
6960:
6938:
6936:
6935:
6930:
6910:
6909:
6881:
6879:
6878:
6873:
6853:
6852:
6823:
6817:
6811:
6809:
6808:
6803:
6801:
6800:
6795:
6790:
6779:
6661:Projective range
6646:
6644:
6643:
6638:
6633:
6628:
6623:
6610:
6608:
6607:
6602:
6600:
6599:
6594:
6589:
6573:
6571:
6570:
6565:
6563:
6551:
6549:
6548:
6543:
6538:
6530:
6515:
6513:
6512:
6507:
6502:
6497:
6492:
6480:
6478:
6477:
6472:
6470:
6469:
6464:
6459:
6443:
6441:
6440:
6435:
6420:
6418:
6417:
6412:
6407:
6395:
6393:
6392:
6387:
6372:
6370:
6369:
6364:
6359:
6335:
6333:
6332:
6327:
6322:
6321:
6316:
6311:
6301:
6299:
6298:
6293:
6291:
6273:
6271:
6270:
6265:
6260:
6259:
6254:
6249:
6239:
6237:
6236:
6231:
6226:
6214:
6212:
6211:
6206:
6191:
6189:
6188:
6183:
6178:
6167:
6164:
6156:
6152:
6142:
6134:
6111:
6109:
6108:
6103:
6081:if they are not
6080:
6078:
6077:
6072:
6057:
6055:
6054:
6049:
6033:
6031:
6030:
6025:
6020:
6019:
6014:
6009:
5999:
5997:
5996:
5991:
5989:
5988:
5983:
5978:
5968:
5966:
5965:
5960:
5958:
5957:
5952:
5947:
5933:
5927:
5921:
5897:
5895:
5894:
5889:
5865:
5859:
5853:
5847:
5845:
5844:
5839:
5818:
5812:
5806:
5804:
5803:
5798:
5780:
5771:
5769:
5768:
5763:
5761:
5760:
5755:
5750:
5725:
5723:
5722:
5717:
5712:
5711:
5706:
5701:
5682:
5680:
5679:
5674:
5654:
5639:
5614:
5608:
5602:
5589:
5587:
5586:
5581:
5576:
5575:
5570:
5565:
5552:
5551:
5546:
5541:
5535:
5534:
5529:
5524:
5498:
5496:
5495:
5490:
5455:
5453:
5452:
5447:
5410:
5408:
5407:
5402:
5372:
5370:
5369:
5364:
5292:
5290:
5289:
5284:
5282:
5281:
5276:
5271:
5259:
5258:
5253:
5248:
5236:
5235:
5230:
5225:
5213:
5212:
5207:
5202:
5196:
5195:
5190:
5185:
5167:
5165:
5164:
5159:
5138:
5136:
5135:
5130:
5096:
5094:
5093:
5088:
5086:
5085:
5080:
5075:
5051:
5049:
5048:
5043:
5032:
5031:
5014:
5007:
4982:
4980:
4979:
4974:
4966:
4951:
4950:
4949:
4921:
4919:
4918:
4913:
4902:
4901:
4884:
4877:
4852:
4850:
4849:
4844:
4836:
4821:
4820:
4819:
4791:
4789:
4788:
4783:
4772:
4771:
4754:
4747:
4725:
4723:
4722:
4717:
4709:
4694:
4671:
4669:
4668:
4663:
4655:
4640:
4615:
4613:
4612:
4607:
4599:
4584:
4583:
4582:
4554:
4552:
4551:
4546:
4538:
4523:
4498:
4496:
4495:
4490:
4482:
4467:
4466:
4465:
4437:
4435:
4434:
4429:
4421:
4406:
4384:
4382:
4381:
4376:
4368:
4353:
4328:
4326:
4325:
4320:
4315:
4288:
4286:
4285:
4280:
4278:
4262:
4260:
4259:
4254:
4246:
4231:
4230:
4229:
4202:
4200:
4199:
4194:
4186:
4171:
4170:
4169:
4142:
4140:
4139:
4134:
4126:
4111:
4087:
4085:
4084:
4079:
4049:
4047:
4046:
4041:
4004:
4002:
4001:
3996:
3991:
3987:
3980:
3965:
3964:
3963:
3931:
3916:
3915:
3914:
3857:
3855:
3854:
3849:
3822:
3820:
3819:
3814:
3777:
3775:
3774:
3769:
3761:
3746:
3713:
3692:
3690:
3689:
3684:
3682:
3681:
3676:
3671:
3655:
3653:
3652:
3647:
3642:
3641:
3636:
3631:
3616:
3614:
3613:
3608:
3603:
3602:
3597:
3592:
3586:
3585:
3580:
3575:
3545:
3541:
3539:
3538:
3533:
3509:
3507:
3506:
3501:
3480:
3472:
3466:
3464:
3463:
3458:
3434:
3430:
3428:
3427:
3422:
3404:
3400:
3396:
3390:
3388:
3387:
3382:
3358:
3354:
3352:
3351:
3346:
3328:
3324:
3320:
3314:
3310:
3304:
3298:
3288:
3281:
3279:
3278:
3273:
3271:
3270:
3265:
3260:
3244:
3242:
3241:
3236:
3234:
3233:
3228:
3223:
3199:
3197:
3196:
3191:
3189:
3188:
3183:
3178:
3156:
3152:
3145:
3143:
3142:
3137:
3131:
3127:
3126:
3118:
3113:
3105:
3095:
3087:
3049:
3047:
3046:
3041:
3039:
3038:
3033:
3028:
3006:
3004:
3003:
2998:
2996:
2984:
2982:
2981:
2976:
2974:
2973:
2968:
2963:
2943:
2941:
2940:
2935:
2930:
2904:
2902:
2901:
2896:
2875:
2873:
2872:
2867:
2865:
2841:
2839:
2838:
2833:
2816:
2766:
2721:
2719:
2718:
2713:
2711:
2695:
2693:
2692:
2687:
2685:
2684:
2679:
2674:
2653:
2651:
2650:
2645:
2643:
2642:
2637:
2632:
2618:
2616:
2615:
2610:
2608:
2607:
2602:
2597:
2578:
2576:
2575:
2570:
2568:
2548:
2544:
2509:
2505:
2471:
2432:
2428:
2394:
2344:
2317:
2313:
2267:
2240:
2236:
2196:
2169:
2165:
2136:
2134:
2133:
2128:
2114:
2090:
2088:
2087:
2082:
2080:
2079:
2074:
2069:
2047:
2045:
2044:
2039:
2037:
2036:
2031:
2026:
2016:
2014:
2013:
2008:
2006:
1991:
1989:
1988:
1983:
1981:
1910:
1905:
1888:
1871:
1867:
1859:
1777:
1775:
1774:
1769:
1764:
1763:
1758:
1753:
1722:
1720:
1719:
1714:
1712:
1708:
1700:
1505:
1503:
1502:
1497:
1492:
1491:
1486:
1481:
1441:
1439:
1438:
1433:
1431:
1430:
1425:
1420:
1410:
1408:
1407:
1402:
1400:
1399:
1377:
1375:
1374:
1369:
1367:
1360:
1351:
1344:
1335:
1321:
1307:
1293:
1279:
1256:
1254:
1253:
1248:
1246:
1215:
1152:
1113:
1050:
930:
928:
927:
922:
920:
919:
914:
909:
893:
891:
890:
885:
883:
882:
877:
872:
862:
860:
859:
854:
852:
812:
810:
809:
804:
802:
801:
796:
782:linear subspaces
773:The terminology
757:
726:
722:
683:
671:
669:
668:
663:
661:
660:
655:
650:
640:
638:
637:
632:
630:
618:
616:
615:
610:
608:
607:
602:
597:
587:
583:
576:
569:
562:
560:
559:
554:
552:
551:
546:
541:
519:
515:
510:compactification
500:
472:
464:
448:
435:
431:
420:
418:
417:
412:
404:
396:
383:division by zero
377:Dividing by zero
372:
368:
360:
356:
352:
313:
311:
310:
305:
300:
299:
294:
289:
279:
277:
276:
271:
269:
268:
263:
250:
248:
247:
242:
228:
216:
212:
210:
209:
204:
202:
140:
133:
129:
126:
120:
118:
77:
53:
45:
37:
33:
21:
7356:
7355:
7351:
7350:
7349:
7347:
7346:
7345:
7316:
7315:
7314:
7313:
7306:
7291:
7290:
7286:
7277:
7275:
7266:
7265:
7250:
7238:
7237:
7224:
7219:
7214:
7213:
7206:
7202:
7151:
7150:
7148:
7144:
7110:
7109:
7067:
7066:
7035:
7034:
7004:
7003:
6984:
6983:
6941:
6940:
6901:
6884:
6883:
6844:
6827:
6826:
6824:
6820:
6783:
6782:
6780:
6776:
6771:
6749:
6663:
6657:
6617:
6616:
6615:is extended to
6576:
6575:
6554:
6553:
6518:
6517:
6486:
6485:
6452:
6451:
6450:is extended to
6423:
6422:
6398:
6397:
6375:
6374:
6350:
6349:
6304:
6303:
6282:
6281:
6242:
6241:
6217:
6216:
6197:
6196:
6165: for
6132:
6128:
6117:
6116:
6094:
6093:
6060:
6059:
6040:
6039:
6002:
6001:
5971:
5970:
5940:
5939:
5929:
5923:
5904:
5868:
5867:
5861:
5855:
5849:
5821:
5820:
5814:
5808:
5783:
5782:
5776:
5731:
5730:
5688:
5687:
5620:
5619:
5610:
5604:
5603:if and only if
5598:
5511:
5510:
5504:
5466:
5465:
5423:
5422:
5375:
5374:
5343:
5342:
5301:. The limit of
5172:
5171:
5141:
5140:
5115:
5114:
5062:
5061:
5058:
4985:
4984:
4941:
4925:
4924:
4855:
4854:
4811:
4795:
4794:
4728:
4727:
4675:
4674:
4618:
4617:
4574:
4558:
4557:
4501:
4500:
4457:
4441:
4440:
4387:
4386:
4334:
4333:
4294:
4293:
4290:
4269:
4268:
4221:
4205:
4204:
4161:
4145:
4144:
4092:
4091:
4052:
4051:
4026:
4025:
3955:
3943:
3939:
3906:
3890:
3889:
3863:one-sided limit
3825:
3824:
3799:
3798:
3727:
3726:
3709:
3658:
3657:
3618:
3617:
3562:
3561:
3558:
3553:
3543:
3512:
3511:
3483:
3482:
3478:
3470:
3437:
3436:
3432:
3407:
3406:
3402:
3398:
3394:
3361:
3360:
3356:
3331:
3330:
3326:
3322:
3318:
3312:
3306:
3300:
3294:
3286:
3247:
3246:
3210:
3209:
3206:
3171:
3170:
3163:
3154:
3150:
3103:
3099:
3055:
3054:
3009:
3008:
2987:
2986:
2956:
2955:
2949:
2921:
2920:
2878:
2877:
2844:
2843:
2724:
2723:
2702:
2701:
2667:
2666:
2625:
2624:
2590:
2589:
2566:
2565:
2549:
2534:
2530:
2527:
2526:
2510:
2495:
2491:
2488:
2487:
2433:
2418:
2414:
2411:
2410:
2318:
2303:
2299:
2296:
2295:
2241:
2226:
2222:
2219:
2218:
2170:
2155:
2151:
2142:
2141:
2093:
2092:
2062:
2061:
2054:
2019:
2018:
1997:
1996:
1979:
1978:
1953:
1946:
1918:
1912:
1911:
1889:
1885:
1878:
1854:
1847:
1841:
1840:
1812:
1782:
1781:
1728:
1727:
1710:
1709:
1692:
1680:
1679:
1663:
1651:
1650:
1622:
1598:
1597:
1581:
1569:
1568:
1540:
1510:
1509:
1456:
1455:
1448:
1413:
1412:
1391:
1386:
1385:
1365:
1364:
1349:
1348:
1333:
1332:
1319:
1318:
1305:
1304:
1291:
1290:
1271:
1270:
1263:
1244:
1243:
1232:
1219:
1205:
1204:
1193:
1180:
1142:
1141:
1130:
1117:
1103:
1102:
1091:
1078:
1040:
1039:
1028:
1015:
991:
990:
979:
966:
936:
935:
896:
895:
865:
864:
843:
842:
836:
824:
819:
791:
786:
785:
775:projective line
755:
724:
723:, the image is
720:
681:
678:
643:
642:
621:
620:
590:
589:
585:
578:
571:
564:
534:
533:
526:
517:
513:
498:
495:complex numbers
479:
470:
462:
443:
433:
429:
390:
389:
379:
370:
366:
358:
354:
350:
332:absolute values
282:
281:
258:
253:
252:
219:
218:
214:
193:
192:
141:
130:
124:
121:
78:
76:
66:
54:
43:
35:
31:
28:
23:
22:
18:Projected reals
15:
12:
11:
5:
7354:
7352:
7344:
7343:
7338:
7333:
7328:
7318:
7317:
7312:
7311:
7304:
7284:
7248:
7221:
7220:
7218:
7215:
7212:
7211:
7190:
7187:
7184:
7181:
7178:
7174:
7170:
7167:
7164:
7161:
7158:
7142:
7129:
7126:
7123:
7120:
7117:
7097:
7094:
7091:
7088:
7082:
7078:
7054:
7051:
7048:
7045:
7042:
7019:
7015:
6991:
6968:
6964:
6957:
6954:
6951:
6948:
6928:
6925:
6922:
6919:
6916:
6913:
6908:
6904:
6900:
6897:
6894:
6891:
6871:
6868:
6865:
6862:
6859:
6856:
6851:
6847:
6843:
6840:
6837:
6834:
6818:
6798:
6794:
6773:
6772:
6770:
6767:
6766:
6765:
6760:
6755:
6748:
6745:
6694:projectivities
6659:Main article:
6656:
6653:
6636:
6631:
6627:
6597:
6593:
6586:
6583:
6562:
6541:
6536:
6533:
6528:
6525:
6505:
6500:
6496:
6467:
6463:
6433:
6430:
6410:
6406:
6385:
6382:
6362:
6358:
6325:
6319:
6315:
6290:
6263:
6257:
6253:
6229:
6225:
6204:
6193:
6192:
6181:
6177:
6173:
6170:
6162:
6159:
6155:
6151:
6148:
6145:
6140:
6137:
6131:
6127:
6124:
6101:
6070:
6067:
6047:
6023:
6017:
6013:
5986:
5982:
5955:
5951:
5887:
5884:
5881:
5878:
5875:
5837:
5834:
5831:
5828:
5813:is defined at
5796:
5793:
5790:
5773:
5772:
5758:
5754:
5747:
5744:
5741:
5738:
5715:
5709:
5705:
5698:
5695:
5684:
5683:
5672:
5669:
5666:
5663:
5660:
5657:
5653:
5650:
5647:
5644:
5638:
5635:
5632:
5628:
5609:is defined at
5591:
5590:
5579:
5573:
5569:
5562:
5559:
5555:
5549:
5545:
5538:
5532:
5528:
5521:
5518:
5503:
5500:
5488:
5485:
5482:
5479:
5476:
5473:
5445:
5442:
5439:
5436:
5433:
5430:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5362:
5359:
5356:
5353:
5350:
5279:
5275:
5268:
5265:
5262:
5256:
5252:
5245:
5242:
5239:
5233:
5229:
5222:
5219:
5216:
5210:
5206:
5199:
5193:
5189:
5182:
5179:
5157:
5154:
5151:
5148:
5128:
5125:
5122:
5083:
5079:
5072:
5069:
5057:
5054:
5053:
5052:
5041:
5038:
5035:
5030:
5026:
5023:
5020:
5017:
5013:
5006:
5003:
5000:
4997:
4993:
4972:
4969:
4965:
4962:
4959:
4956:
4948:
4944:
4940:
4937:
4933:
4922:
4911:
4908:
4905:
4900:
4896:
4893:
4890:
4887:
4883:
4876:
4873:
4870:
4867:
4863:
4842:
4839:
4835:
4832:
4829:
4826:
4818:
4814:
4810:
4807:
4803:
4792:
4781:
4778:
4775:
4770:
4766:
4763:
4760:
4757:
4753:
4746:
4743:
4740:
4736:
4715:
4712:
4708:
4705:
4702:
4699:
4693:
4690:
4687:
4683:
4672:
4661:
4658:
4654:
4651:
4648:
4645:
4639:
4636:
4633:
4630:
4626:
4605:
4602:
4598:
4595:
4592:
4589:
4581:
4577:
4573:
4570:
4566:
4555:
4544:
4541:
4537:
4534:
4531:
4528:
4522:
4519:
4516:
4513:
4509:
4488:
4485:
4481:
4478:
4475:
4472:
4464:
4460:
4456:
4453:
4449:
4438:
4427:
4424:
4420:
4417:
4414:
4411:
4405:
4402:
4399:
4395:
4374:
4371:
4367:
4364:
4361:
4358:
4352:
4349:
4346:
4342:
4318:
4314:
4310:
4307:
4304:
4301:
4289:
4277:
4265:
4252:
4249:
4245:
4242:
4239:
4236:
4228:
4224:
4220:
4217:
4213:
4192:
4189:
4185:
4182:
4179:
4176:
4168:
4164:
4160:
4157:
4153:
4132:
4129:
4125:
4122:
4119:
4116:
4110:
4107:
4104:
4100:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4039:
4036:
4033:
4006:
4005:
3994:
3990:
3986:
3983:
3979:
3976:
3973:
3970:
3962:
3958:
3954:
3951:
3947:
3942:
3937:
3934:
3930:
3927:
3924:
3921:
3913:
3909:
3905:
3902:
3898:
3847:
3844:
3841:
3838:
3835:
3832:
3812:
3809:
3806:
3779:
3778:
3767:
3764:
3760:
3757:
3754:
3751:
3745:
3742:
3739:
3735:
3679:
3675:
3668:
3665:
3645:
3639:
3635:
3628:
3625:
3606:
3600:
3596:
3589:
3583:
3579:
3572:
3569:
3557:
3554:
3552:
3549:
3548:
3547:
3531:
3528:
3525:
3522:
3519:
3499:
3496:
3493:
3490:
3468:
3456:
3453:
3450:
3447:
3444:
3420:
3417:
3414:
3392:
3380:
3377:
3374:
3371:
3368:
3344:
3341:
3338:
3316:
3311:that contains
3268:
3264:
3257:
3254:
3231:
3227:
3220:
3217:
3205:
3204:Neighbourhoods
3202:
3186:
3182:
3162:
3159:
3147:
3146:
3135:
3130:
3124:
3121:
3116:
3111:
3108:
3102:
3098:
3093:
3090:
3084:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3036:
3032:
3025:
3022:
3019:
3016:
2995:
2971:
2967:
2948:
2945:
2933:
2929:
2894:
2891:
2888:
2885:
2864:
2860:
2857:
2854:
2851:
2831:
2828:
2825:
2822:
2819:
2815:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2765:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2710:
2682:
2678:
2640:
2636:
2605:
2601:
2580:
2579:
2564:
2561:
2558:
2555:
2552:
2550:
2547:
2543:
2540:
2537:
2533:
2529:
2528:
2525:
2522:
2519:
2516:
2513:
2511:
2508:
2504:
2501:
2498:
2494:
2490:
2489:
2486:
2483:
2480:
2477:
2474:
2470:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2434:
2431:
2427:
2424:
2421:
2417:
2413:
2412:
2409:
2406:
2403:
2400:
2397:
2393:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2343:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2319:
2316:
2312:
2309:
2306:
2302:
2298:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2266:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2242:
2239:
2235:
2232:
2229:
2225:
2221:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2195:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2171:
2168:
2164:
2161:
2158:
2154:
2150:
2149:
2126:
2123:
2120:
2117:
2113:
2109:
2106:
2103:
2100:
2077:
2073:
2053:
2050:
2034:
2030:
2005:
1993:
1992:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1954:
1950:
1947:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1919:
1917:
1914:
1913:
1908:
1904:
1901:
1898:
1895:
1892:
1886:
1882:
1879:
1877:
1874:
1870:
1865:
1862:
1857:
1853:
1850:
1848:
1846:
1843:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1813:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1789:
1767:
1761:
1757:
1750:
1747:
1744:
1741:
1738:
1735:
1724:
1723:
1706:
1703:
1698:
1695:
1693:
1691:
1688:
1685:
1682:
1681:
1678:
1675:
1672:
1669:
1666:
1664:
1662:
1659:
1656:
1653:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1623:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1599:
1596:
1593:
1590:
1587:
1584:
1582:
1580:
1577:
1574:
1571:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1541:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1517:
1495:
1489:
1485:
1478:
1475:
1472:
1469:
1466:
1463:
1447:
1444:
1428:
1424:
1398:
1394:
1379:
1378:
1363:
1359:
1355:
1352:
1350:
1347:
1343:
1339:
1336:
1334:
1331:
1328:
1325:
1322:
1320:
1317:
1314:
1311:
1308:
1306:
1303:
1300:
1297:
1294:
1292:
1289:
1286:
1283:
1280:
1278:
1262:
1259:
1258:
1257:
1242:
1239:
1236:
1233:
1231:
1228:
1225:
1222:
1220:
1218:
1214:
1210:
1207:
1206:
1203:
1200:
1197:
1194:
1192:
1189:
1186:
1183:
1181:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1151:
1147:
1144:
1143:
1140:
1137:
1134:
1131:
1129:
1126:
1123:
1120:
1118:
1116:
1112:
1108:
1105:
1104:
1101:
1098:
1095:
1092:
1090:
1087:
1084:
1081:
1079:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1049:
1045:
1042:
1041:
1038:
1035:
1032:
1029:
1027:
1024:
1021:
1018:
1016:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
992:
989:
986:
983:
980:
978:
975:
972:
969:
967:
965:
962:
959:
956:
953:
950:
947:
944:
943:
917:
913:
906:
903:
880:
876:
851:
835:
832:
823:
820:
818:
815:
800:
795:
770:is invariant.
677:
674:
658:
654:
629:
605:
601:
549:
545:
525:
522:
491:Riemann sphere
478:
475:
451:total function
422:
421:
410:
407:
402:
399:
378:
375:
373:are distinct.
303:
297:
293:
267:
262:
240:
237:
234:
231:
227:
201:
143:
142:
57:
55:
48:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7353:
7342:
7339:
7337:
7334:
7332:
7329:
7327:
7326:Real analysis
7324:
7323:
7321:
7307:
7301:
7297:
7296:
7288:
7285:
7274:
7270:
7263:
7261:
7259:
7257:
7255:
7253:
7249:
7244:
7243:
7242:PG MTM 201 B1
7235:
7233:
7231:
7229:
7227:
7223:
7216:
7185:
7182:
7179:
7172:
7165:
7162:
7159:
7146:
7143:
7124:
7121:
7118:
7092:
7080:
7049:
7046:
7043:
7017:
6966:
6955:
6952:
6949:
6946:
6923:
6920:
6917:
6911:
6906:
6898:
6895:
6892:
6866:
6863:
6860:
6854:
6849:
6841:
6838:
6835:
6822:
6819:
6815:
6796:
6778:
6775:
6768:
6764:
6761:
6759:
6756:
6754:
6751:
6750:
6746:
6744:
6742:
6741:reciprocation
6738:
6734:
6730:
6726:
6721:
6719:
6717:
6711:
6707:
6703:
6699:
6698:automorphisms
6695:
6690:
6688:
6684:
6680:
6676:
6672:
6668:
6662:
6654:
6652:
6650:
6634:
6614:
6595:
6584:
6539:
6534:
6531:
6523:
6503:
6484:
6465:
6449:
6444:
6431:
6408:
6383:
6360:
6347:
6343:
6339:
6323:
6317:
6279:
6274:
6261:
6255:
6227:
6202:
6179:
6171:
6168:
6157:
6153:
6149:
6146:
6143:
6138:
6135:
6129:
6125:
6122:
6115:
6114:
6113:
6099:
6091:
6088:Also, if the
6086:
6084:
6068:
6037:
6021:
6015:
5984:
5953:
5937:
5932:
5926:
5919:
5915:
5911:
5907:
5903:
5898:
5885:
5879:
5873:
5864:
5858:
5852:
5832:
5826:
5817:
5811:
5794:
5791:
5788:
5779:
5756:
5742:
5739:
5736:
5729:
5728:
5727:
5726:the function
5713:
5707:
5696:
5693:
5670:
5664:
5658:
5655:
5648:
5642:
5636:
5630:
5618:
5617:
5616:
5613:
5607:
5601:
5596:
5577:
5571:
5560:
5557:
5553:
5547:
5530:
5519:
5516:
5509:
5508:
5507:
5506:The function
5501:
5499:
5486:
5480:
5474:
5471:
5463:
5459:
5443:
5437:
5431:
5428:
5420:
5416:
5411:
5398:
5395:
5392:
5386:
5380:
5360:
5357:
5354:
5351:
5348:
5340:
5336:
5332:
5328:
5324:
5320:
5316:
5312:
5308:
5304:
5300:
5296:
5277:
5266:
5263:
5260:
5254:
5243:
5240:
5237:
5231:
5220:
5217:
5214:
5208:
5191:
5180:
5177:
5168:
5155:
5152:
5149:
5146:
5126:
5123:
5120:
5112:
5108:
5104:
5100:
5081:
5070:
5067:
5055:
5036:
5033:
5021:
5015:
5001:
4995:
4967:
4960:
4954:
4946:
4935:
4923:
4906:
4903:
4891:
4885:
4871:
4865:
4837:
4830:
4824:
4816:
4805:
4793:
4776:
4773:
4761:
4755:
4744:
4738:
4710:
4703:
4697:
4691:
4685:
4673:
4659:
4656:
4649:
4643:
4634:
4628:
4603:
4600:
4593:
4587:
4579:
4568:
4556:
4542:
4539:
4532:
4526:
4517:
4511:
4486:
4483:
4476:
4470:
4462:
4451:
4439:
4425:
4422:
4415:
4409:
4403:
4397:
4372:
4369:
4362:
4356:
4350:
4344:
4332:
4331:
4330:
4316:
4308:
4305:
4302:
4299:
4264:
4250:
4247:
4240:
4234:
4226:
4222:
4215:
4190:
4187:
4180:
4174:
4166:
4162:
4155:
4130:
4127:
4120:
4114:
4108:
4102:
4088:
4075:
4072:
4069:
4063:
4057:
4037:
4034:
4031:
4023:
4019:
4015:
4011:
3992:
3988:
3984:
3981:
3974:
3968:
3960:
3956:
3949:
3940:
3935:
3932:
3925:
3919:
3911:
3907:
3900:
3888:
3887:
3886:
3884:
3880:
3876:
3872:
3868:
3864:
3859:
3845:
3842:
3836:
3830:
3810:
3807:
3804:
3796:
3792:
3788:
3784:
3765:
3762:
3755:
3749:
3743:
3737:
3725:
3724:
3723:
3721:
3717:
3712:
3707:
3703:
3699:
3694:
3677:
3666:
3663:
3643:
3637:
3626:
3623:
3604:
3598:
3581:
3570:
3567:
3555:
3550:
3526:
3520:
3517:
3497:
3494:
3491:
3488:
3476:
3469:
3451:
3448:
3445:
3418:
3415:
3412:
3393:
3375:
3372:
3369:
3342:
3339:
3336:
3317:
3309:
3303:
3297:
3292:
3291:neighbourhood
3285:
3284:
3283:
3266:
3255:
3252:
3229:
3218:
3215:
3203:
3201:
3184:
3168:
3165:The tools of
3160:
3158:
3133:
3128:
3122:
3119:
3114:
3109:
3106:
3100:
3096:
3091:
3088:
3075:
3072:
3069:
3063:
3060:
3053:
3052:
3051:
3034:
3023:
3020:
3017:
3014:
2969:
2953:
2946:
2944:
2931:
2918:
2914:
2910:
2905:
2892:
2889:
2886:
2883:
2858:
2855:
2852:
2849:
2826:
2823:
2820:
2817:
2809:
2806:
2803:
2800:
2794:
2782:
2776:
2773:
2770:
2767:
2759:
2756:
2753:
2750:
2744:
2738:
2735:
2732:
2699:
2680:
2664:
2660:
2655:
2638:
2622:
2603:
2587:
2585:
2553:
2551:
2545:
2538:
2531:
2520:
2514:
2512:
2506:
2502:
2499:
2496:
2492:
2481:
2478:
2475:
2472:
2464:
2461:
2458:
2455:
2449:
2437:
2435:
2429:
2425:
2422:
2415:
2404:
2401:
2398:
2395:
2387:
2384:
2381:
2378:
2372:
2360:
2354:
2351:
2348:
2345:
2337:
2334:
2331:
2328:
2322:
2320:
2314:
2310:
2307:
2304:
2300:
2283:
2277:
2274:
2271:
2268:
2260:
2257:
2254:
2251:
2245:
2243:
2237:
2230:
2227:
2223:
2212:
2209:
2206:
2203:
2200:
2197:
2189:
2186:
2183:
2180:
2174:
2172:
2166:
2162:
2159:
2156:
2152:
2140:
2139:
2138:
2124:
2121:
2118:
2115:
2107:
2104:
2101:
2098:
2075:
2059:
2051:
2049:
2032:
1975:
1972:
1966:
1963:
1960:
1955:
1948:
1943:
1940:
1934:
1931:
1928:
1922:
1920:
1915:
1906:
1899:
1896:
1893:
1880:
1875:
1872:
1868:
1863:
1860:
1855:
1851:
1849:
1844:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1814:
1806:
1803:
1800:
1794:
1791:
1780:
1779:
1778:
1765:
1759:
1748:
1745:
1742:
1739:
1736:
1733:
1704:
1701:
1696:
1694:
1686:
1683:
1676:
1673:
1670:
1667:
1665:
1660:
1657:
1654:
1644:
1641:
1638:
1632:
1629:
1626:
1624:
1619:
1616:
1610:
1607:
1604:
1594:
1591:
1588:
1585:
1583:
1578:
1575:
1572:
1562:
1559:
1556:
1550:
1547:
1544:
1542:
1537:
1534:
1528:
1525:
1522:
1508:
1507:
1506:
1493:
1487:
1476:
1473:
1470:
1467:
1464:
1461:
1453:
1445:
1443:
1426:
1396:
1392:
1384:
1361:
1357:
1353:
1341:
1326:
1323:
1315:
1312:
1298:
1284:
1269:
1268:
1267:
1260:
1240:
1237:
1234:
1229:
1226:
1223:
1221:
1216:
1212:
1208:
1201:
1198:
1195:
1190:
1184:
1182:
1177:
1174:
1168:
1162:
1159:
1156:
1153:
1149:
1145:
1135:
1132:
1127:
1121:
1119:
1114:
1110:
1096:
1093:
1088:
1085:
1082:
1080:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1047:
1043:
1033:
1030:
1025:
1019:
1017:
1012:
1009:
1003:
997:
994:
984:
981:
976:
970:
968:
963:
960:
954:
948:
945:
934:
933:
932:
915:
904:
901:
878:
841:
833:
831:
829:
821:
816:
814:
798:
783:
780:
776:
771:
769:
765:
761:
758:that is, the
753:
749:
745:
741:
737:
733:
728:
718:
714:
710:
706:
703:
699:
695:
691:
687:
675:
673:
656:
603:
581:
574:
567:
547:
531:
523:
521:
511:
507:
502:
496:
492:
488:
484:
476:
474:
468:
460:
456:
452:
447:
442:
439:
436:, making the
427:
405:
400:
397:
388:
387:
386:
384:
376:
374:
364:
348:
343:
341:
337:
333:
329:
325:
321:
317:
301:
295:
265:
229:
190:
186:
182:
178:
174:
170:
169:real analysis
162:
158:
154:
149:
139:
136:
128:
117:
114:
110:
107:
103:
100:
96:
93:
89:
86: –
85:
81:
80:Find sources:
74:
70:
64:
63:
58:This article
56:
52:
47:
46:
41:
19:
7294:
7287:
7276:. Retrieved
7272:
7241:
7145:
6821:
6814:Wheel theory
6777:
6763:Wheel theory
6733:fixed points
6728:
6722:
6715:
6714:PGL(2,
6702:homographies
6691:
6664:
6445:
6275:
6194:
6087:
5930:
5924:
5917:
5913:
5909:
5905:
5899:
5862:
5856:
5850:
5815:
5809:
5777:
5774:
5685:
5611:
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5341:, such that
5338:
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5318:
5314:
5310:
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5169:
5110:
5106:
5098:
5059:
4291:
4089:
4024:, such that
4021:
4017:
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3866:
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3797:, such that
3794:
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2950:
2909:homeomorphic
2906:
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1449:
1380:
1264:
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713:group action
694:homeomorphic
685:
679:
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572:
565:
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503:
487:real numbers
480:
445:
425:
424:for nonzero
423:
380:
344:
189:real numbers
172:
166:
131:
125:January 2023
122:
112:
105:
98:
91:
79:
67:Please help
62:verification
59:
6725:involutions
5936:polynomials
5458:restriction
5313:approaches
5103:limit point
3877:approaches
3714:approaches
2954:extends to
779:dimensional
768:cross-ratio
711:on it. The
684:is a point
365:, in which
7320:Categories
7278:2023-01-22
7217:References
6649:arctangent
5595:continuous
5502:Continuity
5139:such that
3885:, denoted
3722:, denoted
3405:such that
3329:such that
2913:metrizable
2876:such that
750:to 1, and
702:invertible
692:, in fact
438:reciprocal
336:increasing
95:newspapers
7201:contains
7087:∖
7081:^
7018:^
6990:∅
6967:^
6956:∈
6907:∁
6850:∁
6797:^
6665:When the
6630:¯
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6527:↦
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6429:∞
6381:∞
6318:^
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6161:∞
6150:π
6136:π
6126:
6092:function
6066:∞
6046:∞
6016:^
5985:^
5954:^
5854:tends to
5792:∈
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5232:^
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5005:∞
4999:→
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4809:→
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4742:→
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4689:→
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4632:→
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4455:→
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4309:∈
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3808:∈
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3678:^
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3638:^
3627:∈
3599:^
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3582:^
3521:∪
3416:≠
3340:≠
3267:^
3256:⊆
3230:^
3219:∈
3185:^
3097:∈
3083:⟺
3064:∈
3035:^
3024:∈
2970:^
2859:∈
2810:∈
2804:∣
2795:∪
2789:∞
2783:∪
2760:∈
2754:∣
2681:^
2661:define a
2639:^
2621:empty set
2604:^
2560:∞
2542:∞
2536:∞
2479:≤
2465:∈
2459:∣
2450:∪
2444:∞
2420:∞
2402:≤
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2373:∪
2367:∞
2361:∪
2352:≤
2338:∈
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2290:∞
2284:∪
2275:≤
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2255:∣
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2210:≤
2204:≤
2190:∈
2184:∣
2108:∈
2076:^
2033:^
1964:−
1941:−
1897:⋅
1873:⋅
1835:⋅
1823:⋅
1795:⋅
1760:^
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1690:∞
1687:⋅
1674:⋅
1658:⋅
1642:⋅
1633:⋅
1617:⋅
1608:⋅
1488:^
1477:∈
1427:^
1346:∞
1338:∞
1330:∞
1327:⋅
1313:⋅
1310:∞
1302:∞
1299:−
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1282:∞
1238:≠
1199:≠
1188:∞
1175:⋅
1172:∞
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1136:≠
1125:∞
1107:∞
1100:∞
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1073:⋅
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1052:∞
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1034:≠
1023:∞
1010:−
1007:∞
1001:∞
998:−
988:∞
985:≠
974:∞
958:∞
952:∞
916:^
905:∈
879:^
657:^
604:^
548:^
434:1 / ∞ = 0
430:1 / 0 = ∞
409:∞
340:unbounded
326:of every
296:^
266:∗
236:∞
230:∪
181:real line
7341:Infinity
6939:for all
6747:See also
6737:negation
6731:has two
6687:midpoint
6448:codomain
6340:and all
6083:constant
5922:, where
5860:through
5456:and the
5373:implies
5317:through
5305: (
4050:implies
3869: (
3823:implies
3704: (
3492:∉
3167:calculus
3161:Calculus
2842:for all
2663:topology
2619:and the
2058:interval
705:matrices
676:Geometry
584:. Since
577:or that
441:function
328:sequence
6090:tangent
5097:. Then
2698:bounded
187:of the
179:of the
109:scholar
7302:
6613:domain
5900:Every
3551:Limits
2917:metric
840:subset
828:limits
746:to 0,
707:has a
582:< ∞
575:> ∞
459:binary
171:, the
153:circle
111:
104:
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7207:0 / 0
6769:Notes
6276:Many
6195:then
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5101:is a
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3698:limit
3481:, if
3473:is a
3299:, if
3289:is a
2985:from
760:group
530:order
524:Order
483:field
467:slope
463:0 ⋅ ∞
455:field
324:limit
116:JSTOR
102:books
7300:ISBN
7002:and
6882:and
6739:and
6727:. A
6706:ring
6346:sine
5934:are
5928:and
5615:and
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5060:Let
4203:and
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2122:<
1381:The
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528:The
516:and
444:1 /
432:and
369:and
357:and
338:and
334:are
320:ends
88:news
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754:to
568:≠ ∞
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280:or
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302:.
292:R
261:R
239:}
233:{
226:R
215:∞
200:R
163:.
138:)
132:(
127:)
123:(
113:·
106:·
99:·
92:·
65:.
42:.
20:)
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