2249:
1311:
745:
2868:
816:
225:
933:
377:
2627:
2568:
1474:
1306:{\displaystyle {\begin{aligned}f_{s}(0)&=\lim _{h\to 0}{\frac {f(0+h)-f(0-h)}{2h}}=\lim _{h\to 0}{\frac {f(h)-f(-h)}{2h}}\\&=\lim _{h\to 0}{\frac {1/h^{2}-1/(-h)^{2}}{2h}}=\lim _{h\to 0}{\frac {1/h^{2}-1/h^{2}}{2h}}=\lim _{h\to 0}{\frac {0}{2h}}=0.\end{aligned}}}
740:{\displaystyle {\begin{aligned}f_{s}(0)&=\lim _{h\to 0}{\frac {f(0+h)-f(0-h)}{2h}}=\lim _{h\to 0}{\frac {f(h)-f(-h)}{2h}}\\&=\lim _{h\to 0}{\frac {|h|-|{-h}|}{2h}}\\&=\lim _{h\to 0}{\frac {|h|-|h|}{2h}}\\&=\lim _{h\to 0}{\frac {0}{2h}}=0.\\\end{aligned}}}
2863:{\displaystyle \lim _{h\to 0}{\frac {\operatorname {sgn}(0+h)-2\operatorname {sgn}(0)+\operatorname {sgn}(0-h)}{h^{2}}}=\lim _{h\to 0}{\frac {\operatorname {sgn}(h)-2\cdot 0+(-\operatorname {sgn}(h))}{h^{2}}}=\lim _{h\to 0}{\frac {0}{h^{2}}}=0.}
2449:
1386:
2402:
2086:
938:
2410:
exists, then the second symmetric derivative exists and is equal to it. The second symmetric derivative may exist, however, even when the (ordinary) second derivative does not. As example, consider the
133:
1680:
1538:
382:
2228:
2444:
1502:
902:
346:
310:
804:
Note that in this example both the left and right derivatives at 0 exist, but they are unequal (one is −1, while the other is +1); their average is 0, as expected.
2622:
2596:
1365:
1339:
928:
799:
773:
372:
185:, but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better
2298:
1983:
159:(in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true. A well-known counterexample is the
49:
2563:{\displaystyle \operatorname {sgn}(x)={\begin{cases}-1&{\text{if }}x<0,\\0&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}}
3106:
3082:
3052:
3018:
2985:
2925:
2236:
and its symmetric derivative is also continuous (thus has the
Darboux property), then the function is differentiable in the usual sense.
2147:
1469:{\displaystyle f(x)={\begin{cases}1,&{\text{if }}x{\text{ is rational}}\\0,&{\text{if }}x{\text{ is irrational}}\end{cases}}}
2957:
2893:
2888:
3161:
1367:
due to discontinuity in the curve there. Furthermore, neither the left nor the right derivative is finite at 0, i.e. this is an
1597:
775:
and is equal to zero, even though its ordinary derivative does not exist at that point (due to a "sharp" turn in the curve at
3176:
1507:
3154:
1690:
but for the symmetric derivative was established in 1967 by C. E. Aull, who named it quasi-Rolle theorem. If
3149:
2878:
1593:, the mean-value theorem would mandate that there exist a point where the (symmetric) derivative takes the value
197:
186:
2417:
1368:
156:
3144:
1587:
36:
1479:
3072:
2476:
1410:
2233:
1576:
228:
140:
3162:
Approximating the
Derivative by the Symmetric Difference Quotient (Wolfram Demonstrations Project)
857:
1557:
1380:
208:
1687:
204:
3102:
3078:
3048:
3014:
2981:
2953:
2921:
2407:
1545:
315:
270:
17:
3125:
2125:
2128:, then the (form of the) regular mean-value theorem (of Lagrange) holds, i.e. there exists
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1541:
193:
3010:
3003:
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2575:
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1318:
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hold for the symmetric derivative; some similar but weaker statements have been proved.
264:
160:
2248:
1793:. A lemma also established by Aull as a stepping stone to this theorem states that if
3170:
2883:
2572:
The sign function is not continuous at zero, and therefore the second derivative for
2412:
1712:
3129:
1858:
where the symmetric derivative is non-negative, or with the notation used above,
2106:
28:
815:
224:
40:
2279:
The notion generalizes to higher-order symmetric derivatives and also to
749:
Hence the symmetric derivative of the absolute value function exists at
2397:{\displaystyle \lim _{h\to 0}{\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.}
2081:{\displaystyle f_{s}(x)\leq {\frac {f(b)-f(a)}{b-a}}\leq f_{s}(y).}
814:
223:
1560:(of Lagrange). As a counterexample, the symmetric derivative of
843:. The symmetric derivative, however, exists for the function at
252:. The symmetric derivative, however, exists for the function at
2598:
does not exist. But the second symmetric derivative exists for
2243:
1315:
Again, for this function the symmetric derivative exists at
2556:
2105:
on an interval containing 0 predicts that the slope of any
1928:
for a symmetrically differentiable function states that if
1462:
128:{\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x-h)}{2h}}.}
1675:{\displaystyle {\frac {|2|-|-1|}{2-(-1)}}={\frac {1}{3}}}
836:. The function hence possesses no ordinary derivative at
245:. The function hence possesses no ordinary derivative at
2260:
1586:
can have a wider range of slopes; for instance, on the
1533:{\displaystyle x\in \mathbb {R} \setminus \mathbb {Q} }
231:
of the absolute value function. Note the sharp turn at
137:
The expression under the limit is sometimes called the
3116:
Aull, C. E. (1967). "The first symmetric derivative".
1946:
and symmetrically differentiable on the open interval
1811:
and symmetrically differentiable on the open interval
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The symmetric derivative at a given point equals the
52:
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As an application, the quasi-mean-value theorem for
238:, leading to non-differentiability of the curve at
3002:
2862:
2616:
2590:
2562:
2438:
2396:
2223:{\displaystyle f_{s}(z)={\frac {f(b)-f(a)}{b-a}}.}
2222:
2080:
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1532:
1496:
1468:
1359:
1341:, while its ordinary derivative does not exist at
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1305:
922:
896:
793:
767:
739:
366:
340:
304:
152:if its symmetric derivative exists at that point.
127:
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3034:
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3030:
1556:The symmetric derivative does not obey the usual
1504:, but is not symmetrically differentiable at any
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1110:
1040:
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54:
3077:. Cambridge University Press. pp. 22–23.
2295:The second symmetric derivative is defined as
3066:
3064:
200:at that point, if the latter two both exist.
8:
3045:Mean Value Theorems and Functional Equations
2911:
2909:
3005:Barron's how to Prepare for the AP Calculus
1540:; i.e. the symmetric derivative exists at
2976:Peter D. Lax; Maria Shea Terrell (2013).
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487:
427:
415:
389:
381:
379:
353:
348:for the symmetric derivative, we have at
323:
317:
297:
289:
272:
187:numerical approximation of the derivative
69:
57:
51:
3043:Sahoo, Prasanna; Riedel, Thomas (1998).
3009:. Barron's Educational Series. pp.
1711:and symmetrically differentiable on the
3101:(2nd ed.). CRC Press. p. 34.
3001:Shirley O. Hockett; David Bock (2005).
2905:
1522:
3047:. World Scientific. pp. 188–192.
2950:Symmetric Properties of Real Functions
2439:{\displaystyle \operatorname {sgn}(x)}
1934:is continuous on the closed interval
1799:is continuous on the closed interval
7:
1476:has a symmetric derivative at every
189:than the usual difference quotient.
2232:As a consequence, if a function is
3099:Strange Functions in Real Analysis
2918:More Calculus of a Single Variable
25:
2894:Symmetrically continuous function
2889:Generalizations of the derivative
1497:{\displaystyle x\in \mathbb {Q} }
178:, which is not differentiable at
2247:
1686:A theorem somewhat analogous to
2291:The second symmetric derivative
2118:If the symmetric derivative of
3130:10.1080/00029890.1967.12000020
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2781:
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2194:
2185:
2179:
2167:
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2072:
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2036:
2030:
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2015:
2003:
1997:
1745:, then there exist two points
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108:
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61:
1:
18:Symmetric difference quotient
3097:A. B. Kharazishvili (2005).
1891:, then there exists a point
897:{\displaystyle f(x)=1/x^{2}}
829:. Note the discontinuity at
146:symmetrically differentiable
3150:Encyclopedia of Mathematics
2879:Central differencing scheme
1842:, then there exist a point
220:The absolute value function
144:. A function is said to be
3193:
2978:Calculus With Applications
2948:Thomson, Brian S. (1994).
198:left and right derivatives
39:generalizing the ordinary
2980:. Springer. p. 213.
2920:. Springer. p. 173.
2916:Peter R. Mercer (2014).
1926:quasi-mean-value theorem
1552:Quasi-mean-value theorem
341:{\displaystyle f_{s}(x)}
305:{\displaystyle f(x)=|x|}
1369:essential discontinuity
3145:"Symmetric derivative"
2864:
2618:
2592:
2564:
2446:, which is defined by
2440:
2398:
2224:
2082:
1676:
1534:
1498:
1470:
1375:The Dirichlet function
1361:
1335:
1307:
924:
898:
851:
795:
769:
741:
368:
342:
306:
260:
129:
3177:Differential calculus
2865:
2619:
2593:
2565:
2441:
2399:
2225:
2115:is between −1 and 1.
2083:
1696:is continuous on the
1677:
1535:
1499:
1471:
1362:
1336:
1308:
925:
899:
818:
796:
770:
742:
369:
343:
312:, using the notation
307:
227:
130:
3074:Trigonometric Series
2628:
2602:
2576:
2450:
2418:
2299:
2148:
1984:
1598:
1508:
1480:
1387:
1345:
1319:
934:
908:
858:
779:
753:
378:
352:
316:
271:
50:
33:symmetric derivative
3071:A. Zygmund (2002).
2617:{\displaystyle x=0}
2591:{\displaystyle x=0}
1958:, then there exist
1823:, and additionally
1456: is irrational
1360:{\displaystyle x=0}
1334:{\displaystyle x=0}
923:{\displaystyle x=0}
794:{\displaystyle x=0}
768:{\displaystyle x=0}
367:{\displaystyle x=0}
141:difference quotient
2860:
2836:
2747:
2646:
2614:
2588:
2560:
2555:
2436:
2394:
2317:
2259:. You can help by
2220:
2078:
1872:. Analogously, if
1672:
1582:, but secants for
1558:mean-value theorem
1546:irrational numbers
1530:
1494:
1466:
1461:
1381:Dirichlet function
1357:
1331:
1303:
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1277:
1205:
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426:
364:
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302:
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209:mean-value theorem
125:
68:
46:It is defined as:
3108:978-1-4200-3484-4
3084:978-0-521-89053-3
3054:978-981-02-3544-4
3020:978-0-7641-2382-5
2987:978-1-4614-7946-8
2952:. Marcel Dekker.
2927:978-1-4939-1926-0
2852:
2821:
2816:
2732:
2727:
2631:
2542:
2516:
2490:
2408:second derivative
2389:
2302:
2277:
2276:
2215:
2051:
1670:
1657:
1457:
1449:
1432:
1431: is rational
1424:
1291:
1262:
1257:
1190:
1185:
1109:
1097:
1039:
1034:
967:
854:For the function
725:
696:
684:
627:
615:
553:
541:
483:
478:
411:
155:If a function is
120:
53:
16:(Redirected from
3184:
3158:
3133:
3112:
3089:
3088:
3068:
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2867:
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2861:
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2814:
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2726:
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2716:
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2623:
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2615:
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2595:
2594:
2589:
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2567:
2566:
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2559:
2558:
2543:
2540:
2517:
2514:
2491:
2488:
2445:
2443:
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2390:
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2387:
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2319:
2316:
2285:Euclidean spaces
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2227:
2226:
2221:
2216:
2214:
2203:
2174:
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2159:
2143:
2131:
2126:Darboux property
2123:
2114:
2104:
2087:
2085:
2084:
2079:
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2064:
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2039:
2010:
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1542:rational numbers
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70:
67:
21:
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2531:
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2416:
2415:
2406:If the (usual)
2379:
2320:
2297:
2296:
2293:
2273:
2267:
2264:
2257:needs expansion
2242:
2240:Generalizations
2204:
2175:
2151:
2146:
2145:
2133:
2129:
2119:
2110:
2091:
2056:
2040:
2011:
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1700:
1698:closed interval
1691:
1688:Rolle's theorem
1637:
1603:
1596:
1595:
1594:
1591:[−1, 2]
1590:
1579:
1561:
1554:
1506:
1505:
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1477:
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1459:
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205:Rolle's theorem
194:arithmetic mean
179:
164:
112:
71:
48:
47:
23:
22:
15:
12:
11:
5:
3190:
3188:
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3169:
3168:
3165:
3164:
3159:
3139:
3138:External links
3136:
3135:
3134:
3124:(6): 708–711.
3113:
3107:
3091:
3090:
3083:
3060:
3053:
3026:
3019:
2993:
2986:
2965:
2958:
2933:
2926:
2904:
2903:
2901:
2898:
2897:
2896:
2891:
2886:
2881:
2874:
2871:
2859:
2856:
2849:
2845:
2841:
2834:
2831:
2828:
2824:
2820:
2813:
2809:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2745:
2742:
2739:
2735:
2731:
2724:
2720:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2644:
2641:
2638:
2634:
2613:
2610:
2607:
2587:
2584:
2581:
2557:
2552:
2549:
2546:
2538:
2536:
2533:
2532:
2529:
2526:
2523:
2520:
2512:
2510:
2507:
2506:
2503:
2500:
2497:
2494:
2486:
2484:
2481:
2478:
2477:
2475:
2470:
2467:
2464:
2461:
2458:
2455:
2435:
2432:
2429:
2426:
2423:
2393:
2386:
2382:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2315:
2312:
2309:
2305:
2292:
2289:
2275:
2274:
2254:
2252:
2241:
2238:
2219:
2213:
2210:
2207:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2172:
2169:
2166:
2163:
2158:
2154:
2077:
2074:
2071:
2068:
2063:
2059:
2055:
2049:
2046:
2043:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2008:
2005:
2002:
1999:
1994:
1990:
1912:
1863:
1784:
1770:
1669:
1666:
1661:
1655:
1652:
1649:
1646:
1643:
1640:
1634:
1630:
1627:
1623:
1619:
1615:
1611:
1607:
1553:
1550:
1528:
1524:
1520:
1516:
1513:
1492:
1488:
1485:
1463:
1453:
1445:
1443:
1440:
1437:
1436:
1428:
1420:
1418:
1415:
1412:
1411:
1409:
1404:
1401:
1398:
1395:
1392:
1383:, defined as:
1376:
1373:
1356:
1353:
1350:
1330:
1327:
1324:
1298:
1295:
1289:
1286:
1282:
1275:
1272:
1269:
1265:
1261:
1255:
1252:
1245:
1241:
1236:
1232:
1229:
1224:
1220:
1215:
1211:
1203:
1200:
1197:
1193:
1189:
1183:
1180:
1173:
1169:
1165:
1162:
1159:
1155:
1151:
1148:
1143:
1139:
1134:
1130:
1122:
1119:
1116:
1112:
1108:
1105:
1103:
1101:
1095:
1092:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1052:
1049:
1046:
1042:
1038:
1032:
1029:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
980:
977:
974:
970:
966:
963:
961:
959:
956:
953:
948:
944:
940:
939:
919:
916:
913:
891:
887:
882:
878:
875:
872:
869:
866:
863:
812:
806:
790:
787:
784:
764:
761:
758:
732:
729:
723:
720:
716:
709:
706:
703:
699:
695:
692:
690:
688:
682:
679:
673:
669:
665:
661:
657:
653:
649:
640:
637:
634:
630:
626:
623:
621:
619:
613:
610:
604:
599:
596:
591:
587:
583:
579:
575:
566:
563:
560:
556:
552:
549:
547:
545:
539:
536:
531:
528:
525:
522:
519:
516:
513:
510:
507:
504:
496:
493:
490:
486:
482:
476:
473:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
424:
421:
418:
414:
410:
407:
405:
403:
400:
397:
392:
388:
384:
383:
363:
360:
357:
337:
334:
331:
326:
322:
300:
296:
292:
288:
285:
282:
279:
276:
265:absolute value
221:
218:
216:
213:
161:absolute value
157:differentiable
124:
118:
115:
110:
107:
104:
101:
98:
95:
92:
89:
86:
83:
80:
77:
74:
66:
63:
60:
56:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3189:
3178:
3175:
3174:
3172:
3163:
3160:
3156:
3152:
3151:
3146:
3142:
3141:
3137:
3131:
3127:
3123:
3119:
3118:Am. Math. Mon
3114:
3110:
3104:
3100:
3095:
3094:
3086:
3080:
3076:
3075:
3067:
3065:
3061:
3056:
3050:
3046:
3039:
3037:
3035:
3033:
3031:
3027:
3022:
3016:
3012:
3007:
3006:
2997:
2994:
2989:
2983:
2979:
2972:
2970:
2966:
2961:
2959:0-8247-9230-0
2955:
2951:
2944:
2942:
2940:
2938:
2934:
2929:
2923:
2919:
2912:
2910:
2906:
2899:
2895:
2892:
2890:
2887:
2885:
2884:Density point
2882:
2880:
2877:
2876:
2872:
2870:
2857:
2854:
2847:
2843:
2839:
2832:
2826:
2818:
2811:
2807:
2796:
2790:
2787:
2784:
2778:
2775:
2772:
2769:
2766:
2760:
2754:
2751:
2743:
2737:
2729:
2722:
2718:
2710:
2707:
2704:
2698:
2695:
2692:
2686:
2680:
2677:
2674:
2671:
2665:
2662:
2659:
2653:
2650:
2642:
2636:
2611:
2608:
2605:
2585:
2582:
2579:
2570:
2550:
2547:
2544:
2534:
2527:
2524:
2521:
2518:
2508:
2501:
2498:
2495:
2492:
2482:
2479:
2473:
2468:
2462:
2456:
2453:
2430:
2424:
2421:
2414:
2413:sign function
2409:
2404:
2391:
2384:
2380:
2372:
2369:
2366:
2360:
2357:
2351:
2345:
2342:
2339:
2333:
2330:
2327:
2321:
2313:
2307:
2290:
2288:
2286:
2283:-dimensional
2282:
2271:
2262:
2258:
2255:This section
2253:
2250:
2246:
2245:
2239:
2237:
2235:
2230:
2217:
2211:
2208:
2205:
2197:
2191:
2188:
2182:
2176:
2170:
2164:
2156:
2152:
2141:
2137:
2127:
2122:
2116:
2113:
2108:
2102:
2098:
2094:
2088:
2075:
2069:
2061:
2057:
2053:
2047:
2044:
2041:
2033:
2027:
2024:
2018:
2012:
2006:
2000:
1992:
1988:
1979:
1975:
1971:
1955:
1951:
1943:
1939:
1932:
1927:
1922:
1918:
1911:
1904:
1900:
1888:
1884:
1880:
1876:
1869:
1862:
1855:
1851:
1839:
1835:
1831:
1827:
1820:
1816:
1808:
1804:
1797:
1790:
1783:
1776:
1769:
1762:
1758:
1742:
1738:
1734:
1730:
1723:
1719:
1714:
1713:open interval
1708:
1704:
1699:
1694:
1689:
1684:
1667:
1664:
1659:
1650:
1647:
1641:
1638:
1628:
1625:
1617:
1609:
1589:
1585:
1578:
1572:
1568:
1564:
1559:
1551:
1549:
1547:
1543:
1514:
1511:
1486:
1483:
1451:
1441:
1438:
1426:
1416:
1413:
1407:
1402:
1396:
1390:
1382:
1374:
1372:
1370:
1354:
1351:
1348:
1328:
1325:
1322:
1313:
1296:
1293:
1287:
1284:
1280:
1273:
1267:
1259:
1253:
1250:
1243:
1239:
1234:
1230:
1227:
1222:
1218:
1213:
1209:
1201:
1195:
1187:
1181:
1178:
1171:
1163:
1160:
1153:
1149:
1146:
1141:
1137:
1132:
1128:
1120:
1114:
1106:
1104:
1093:
1090:
1082:
1079:
1073:
1070:
1064:
1058:
1050:
1044:
1036:
1030:
1027:
1019:
1016:
1013:
1007:
1004:
998:
995:
992:
986:
978:
972:
964:
962:
954:
946:
942:
917:
914:
911:
889:
885:
880:
876:
873:
867:
861:
847:
840:
833:
827:
823:
817:
811:
808:The function
807:
805:
802:
788:
785:
782:
762:
759:
756:
747:
730:
727:
721:
718:
714:
707:
701:
693:
691:
680:
677:
667:
659:
651:
638:
632:
624:
622:
611:
608:
597:
594:
585:
577:
564:
558:
550:
548:
537:
534:
526:
523:
517:
514:
508:
502:
494:
488:
480:
474:
471:
463:
460:
457:
451:
448:
442:
439:
436:
430:
422:
416:
408:
406:
398:
390:
386:
361:
358:
355:
332:
324:
320:
294:
286:
280:
274:
266:
256:
249:
242:
235:
230:
226:
219:
214:
212:
210:
206:
201:
199:
195:
190:
188:
182:
175:
171:
167:
162:
158:
153:
151:
147:
143:
142:
135:
122:
116:
113:
105:
102:
99:
93:
90:
84:
81:
78:
72:
64:
58:
44:
42:
38:
34:
30:
19:
3148:
3121:
3117:
3098:
3073:
3044:
3004:
2996:
2977:
2949:
2917:
2571:
2405:
2294:
2280:
2278:
2265:
2261:adding to it
2256:
2231:
2139:
2135:
2120:
2117:
2111:
2100:
2096:
2092:
2089:
1980:
1973:
1969:
1953:
1949:
1941:
1937:
1930:
1925:
1923:
1916:
1909:
1902:
1898:
1886:
1882:
1878:
1874:
1867:
1860:
1853:
1849:
1837:
1833:
1829:
1825:
1818:
1814:
1806:
1802:
1795:
1788:
1781:
1774:
1767:
1760:
1756:
1740:
1736:
1732:
1728:
1721:
1717:
1706:
1702:
1692:
1685:
1583:
1570:
1566:
1562:
1555:
1378:
1314:
853:
845:
838:
831:
825:
821:
809:
803:
748:
262:
254:
247:
240:
233:
202:
191:
180:
173:
169:
165:
154:
149:
145:
138:
136:
45:
32:
26:
1544:but not at
148:at a point
29:mathematics
2900:References
2268:April 2015
2234:continuous
2144:such that
1978:such that
1765:such that
1580:{−1, 0, 1}
139:symmetric
41:derivative
3155:EMS Press
2830:→
2791:
2785:−
2773:⋅
2767:−
2755:
2741:→
2708:−
2699:
2681:
2672:−
2654:
2640:→
2480:−
2457:
2425:
2370:−
2340:−
2311:→
2209:−
2189:−
2054:≤
2045:−
2025:−
2007:≤
1648:−
1642:−
1626:−
1618:−
1523:∖
1515:∈
1487:∈
1271:→
1228:−
1199:→
1161:−
1147:−
1118:→
1080:−
1071:−
1048:→
1017:−
1005:−
976:→
819:Graph of
705:→
660:−
636:→
595:−
586:−
562:→
524:−
515:−
492:→
461:−
449:−
420:→
267:function
163:function
103:−
91:−
62:→
37:operation
3171:Category
2873:See also
2541:if
2515:if
2489:if
2124:has the
1588:interval
1575:has the
1448:if
1423:if
930:we have
263:For the
215:Examples
207:nor the
203:Neither
3157:, 2001
1881:) <
1832:) >
196:of the
3105:
3081:
3051:
3017:
2984:
2956:
2924:
2107:secant
1907:where
1779:, and
1726:, and
35:is an
31:, the
2099:) = |
1944:]
1936:[
1919:) ≤ 0
1870:) ≥ 0
1809:]
1801:[
1791:) ≤ 0
1777:) ≥ 0
1743:) = 0
1709:]
1701:[
1577:image
1569:) = |
904:, at
374:that
229:Graph
172:) = |
3103:ISBN
3079:ISBN
3049:ISBN
3015:ISBN
2982:ISBN
2954:ISBN
2922:ISBN
2548:>
2496:<
1924:The
1735:) =
1379:The
824:= 1/
3126:doi
2823:lim
2788:sgn
2752:sgn
2734:lim
2696:sgn
2678:sgn
2651:sgn
2633:lim
2454:sgn
2422:sgn
2304:lim
2263:.
2132:in
2109:of
1966:in
1895:in
1846:in
1753:in
1264:lim
1192:lim
1111:lim
1041:lim
969:lim
848:= 0
841:= 0
834:= 0
801:).
698:lim
629:lim
555:lim
485:lim
413:lim
257:= 0
250:= 0
243:= 0
236:= 0
183:= 0
55:lim
43:.
27:In
3173::
3153:,
3147:,
3122:74
3120:.
3063:^
3029:^
3013:.
3011:53
2968:^
2936:^
2908:^
2858:0.
2624::
2551:0.
2287:.
2138:,
1972:,
1962:,
1952:,
1940:,
1921:.
1901:,
1852:,
1817:,
1805:,
1759:,
1749:,
1720:,
1705:,
1548:.
1371:.
1297:0.
731:0.
3132:.
3128::
3111:.
3087:.
3057:.
3023:.
2990:.
2962:.
2930:.
2855:=
2848:2
2844:h
2840:0
2833:0
2827:h
2819:=
2812:2
2808:h
2803:)
2800:)
2797:h
2794:(
2782:(
2779:+
2776:0
2770:2
2764:)
2761:h
2758:(
2744:0
2738:h
2730:=
2723:2
2719:h
2714:)
2711:h
2705:0
2702:(
2693:+
2690:)
2687:0
2684:(
2675:2
2669:)
2666:h
2663:+
2660:0
2657:(
2643:0
2637:h
2612:0
2609:=
2606:x
2586:0
2583:=
2580:x
2545:x
2535:1
2528:,
2525:0
2522:=
2519:x
2509:0
2502:,
2499:0
2493:x
2483:1
2474:{
2469:=
2466:)
2463:x
2460:(
2434:)
2431:x
2428:(
2392:.
2385:2
2381:h
2376:)
2373:h
2367:x
2364:(
2361:f
2358:+
2355:)
2352:x
2349:(
2346:f
2343:2
2337:)
2334:h
2331:+
2328:x
2325:(
2322:f
2314:0
2308:h
2281:n
2270:)
2266:(
2218:.
2212:a
2206:b
2201:)
2198:a
2195:(
2192:f
2186:)
2183:b
2180:(
2177:f
2171:=
2168:)
2165:z
2162:(
2157:s
2153:f
2142:)
2140:b
2136:a
2134:(
2130:z
2121:f
2112:f
2103:|
2101:x
2097:x
2095:(
2093:f
2076:.
2073:)
2070:y
2067:(
2062:s
2058:f
2048:a
2042:b
2037:)
2034:a
2031:(
2028:f
2022:)
2019:b
2016:(
2013:f
2004:)
2001:x
1998:(
1993:s
1989:f
1976:)
1974:b
1970:a
1968:(
1964:y
1960:x
1956:)
1954:b
1950:a
1948:(
1942:b
1938:a
1931:f
1917:z
1915:(
1913:s
1910:f
1905:)
1903:b
1899:a
1897:(
1893:z
1889:)
1887:a
1885:(
1883:f
1879:b
1877:(
1875:f
1868:z
1866:(
1864:s
1861:f
1856:)
1854:b
1850:a
1848:(
1844:z
1840:)
1838:a
1836:(
1834:f
1830:b
1828:(
1826:f
1821:)
1819:b
1815:a
1813:(
1807:b
1803:a
1796:f
1789:y
1787:(
1785:s
1782:f
1775:x
1773:(
1771:s
1768:f
1763:)
1761:b
1757:a
1755:(
1751:y
1747:x
1741:b
1739:(
1737:f
1733:a
1731:(
1729:f
1724:)
1722:b
1718:a
1716:(
1707:b
1703:a
1693:f
1682:.
1668:3
1665:1
1660:=
1654:)
1651:1
1645:(
1639:2
1633:|
1629:1
1622:|
1614:|
1610:2
1606:|
1584:f
1573:|
1571:x
1567:x
1565:(
1563:f
1527:Q
1519:R
1512:x
1491:Q
1484:x
1452:x
1442:,
1439:0
1427:x
1417:,
1414:1
1408:{
1403:=
1400:)
1397:x
1394:(
1391:f
1355:0
1352:=
1349:x
1329:0
1326:=
1323:x
1294:=
1288:h
1285:2
1281:0
1274:0
1268:h
1260:=
1254:h
1251:2
1244:2
1240:h
1235:/
1231:1
1223:2
1219:h
1214:/
1210:1
1202:0
1196:h
1188:=
1182:h
1179:2
1172:2
1168:)
1164:h
1158:(
1154:/
1150:1
1142:2
1138:h
1133:/
1129:1
1121:0
1115:h
1107:=
1094:h
1091:2
1086:)
1083:h
1077:(
1074:f
1068:)
1065:h
1062:(
1059:f
1051:0
1045:h
1037:=
1031:h
1028:2
1023:)
1020:h
1014:0
1011:(
1008:f
1002:)
999:h
996:+
993:0
990:(
987:f
979:0
973:h
965:=
958:)
955:0
952:(
947:s
943:f
918:0
915:=
912:x
890:2
886:x
881:/
877:1
874:=
871:)
868:x
865:(
862:f
850:.
846:x
839:x
832:x
826:x
822:y
810:x
789:0
786:=
783:x
763:0
760:=
757:x
728:=
722:h
719:2
715:0
708:0
702:h
694:=
681:h
678:2
672:|
668:h
664:|
656:|
652:h
648:|
639:0
633:h
625:=
612:h
609:2
603:|
598:h
590:|
582:|
578:h
574:|
565:0
559:h
551:=
538:h
535:2
530:)
527:h
521:(
518:f
512:)
509:h
506:(
503:f
495:0
489:h
481:=
475:h
472:2
467:)
464:h
458:0
455:(
452:f
446:)
443:h
440:+
437:0
434:(
431:f
423:0
417:h
409:=
402:)
399:0
396:(
391:s
387:f
362:0
359:=
356:x
336:)
333:x
330:(
325:s
321:f
299:|
295:x
291:|
287:=
284:)
281:x
278:(
275:f
259:.
255:x
248:x
241:x
234:x
181:x
176:|
174:x
170:x
168:(
166:f
150:x
123:.
117:h
114:2
109:)
106:h
100:x
97:(
94:f
88:)
85:h
82:+
79:x
76:(
73:f
65:0
59:h
20:)
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