622:
in order to make the Dini derivatives as "bullet proof" as possible, so that the Dini derivatives are well-defined for almost all functions, even for functions that are not conventionally differentiable. The upshot of Dini's analysis is that a function is differentiable at the point
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381:
233:
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reversed. For only moderately ill-behaved functions, the two extra Dini derivatives aren't needed. For particularly badly behaved functions, if all four Dini derivatives have the same value (
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425:
947:
notation of the Dini derivatives, the plus or minus sign indicates the left- or right-hand limit, and the placement of the sign indicates the
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404:
1322:, each of the Dini derivatives always exist; however, they may take on the values
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20:
1471:
596:
36:
532:{\displaystyle f'_{+}(t,d)=\limsup _{h\to {0+}}{\frac {f(t+hd)-f(t)}{h}}.}
1185:
952:
619:
1189:
948:
615:
549:
376:{\displaystyle f'_{-}(t)=\liminf _{h\to {0+}}{\frac {f(t)-f(t-h)}{h}},}
228:{\displaystyle f'_{+}(t)=\limsup _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}},}
1173:{\displaystyle D^{-}f(t)=\limsup _{h\to {0+}}{\frac {f(t)-f(t-h)}{h}}}
1062:{\displaystyle D_{+}f(t)=\liminf _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}}}
929:{\displaystyle D_{-}f(t)=\liminf _{h\to {0+}}{\frac {f(t)-f(t-h)}{h}}}
818:{\displaystyle D^{+}f(t)=\limsup _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}}}
1449:
Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008).
631:), only if all the Dini derivatives exist, and have the same value.
1470:
This article incorporates material from Dini derivative on
96:{\displaystyle f:{\mathbb {R} }\rightarrow {\mathbb {R} },}
43:, who studied continuous but nondifferentiable functions.
1353: â Fundamental construction of differential calculus
1330:
at times (i.e., the Dini derivatives always exist in the
1355:
Pages displaying short descriptions of redirect targets
1299:{\displaystyle D^{+}f(t)=D_{+}f(t)=D^{-}f(t)=D_{-}f(t)}
1198:
1081:
970:
960:
There are two further Dini derivatives, defined to be
837:
726:
428:
281:
133:
66:
1347: â Mathematical theorem about Dini derivatives
1184:which are the same as the first pair, but with the
1310:is differentiable in the usual sense at the point
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1172:
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817:
531:
375:
227:
95:
1476:Creative Commons Attribution/Share-Alike License
1453:. ClassicalRealAnalysis.com . pp. 301â302.
1108:
997:
864:
753:
461:
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8:
1376:
1374:
614:The functions are defined in terms of the
16:Class of generalisations of the derivative
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1387:(3rd ed.). Upper Saddle River, NJ:
35:) are a class of generalizations of the
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7:
407:, then the upper Dini derivative at
14:
1492:Generalizations of the derivative
1474:, which is licensed under the
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589:, then the Dini derivative at
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1351:Derivative (generalizations)
1434:(2nd ed.). MacMillan.
1418:Encyclopedia of Mathematics
52:upper right-hand derivative
1513:
1411:Lukashenko, T.P. (2001) ,
1381:Khalil, Hassan K. (2002).
50:, which is also called an
39:. They were introduced by
1345:DenjoyâYoungâSaks theorem
1451:Elementary Real Analysis
636:Sometimes the notation
1430:Royden, H. L. (1968).
1360:Semi-differentiability
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930:
819:
533:
377:
229:
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1301:
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252:lower Dini derivative
230:
98:
48:upper Dini derivative
1306:) then the function
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131:
64:
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650:is used instead of
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246:and the limit is a
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56:continuous function
23:and, specifically,
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1016:
941:So when using the
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772:
627:on the real line (
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1460:978-1-4348-4161-2
1441:978-0-02-404150-0
1413:"Dini derivative"
1384:Nonlinear Systems
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413:in the direction
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29:Dini derivatives
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1082:
1077:
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1019:
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838:
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775:
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721:
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661:
660:
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637:
628:
624:
611:
600:
590:
584:
574:
566:
565:
562:
561:
560:
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543:
483:
424:
423:
414:
408:
398:
387:
330:
277:
276:
265:
264:
261:
260:
259:
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248:one-sided limit
239:
182:
129:
128:
124:and defined by
117:
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107:
62:
61:
17:
12:
11:
5:
1510:
1508:
1500:
1499:
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1484:
1483:
1466:
1465:
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1446:
1440:
1427:
1407:
1405:
1404:
1397:
1369:
1367:
1364:
1363:
1362:
1357:
1348:
1340:
1337:
1336:
1335:
1320:extended reals
1295:
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1277:
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1117:
1114:
1110:
1109:lim sup
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1097:
1094:
1089:
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1069:
1056:
1052:
1049:
1046:
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1034:
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1025:
1022:
1013:
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999:
998:lim inf
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986:
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938:
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923:
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913:
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904:
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866:
865:lim inf
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845:
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769:
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755:
754:lim sup
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748:
745:
742:
739:
734:
730:
718:
717:
714:
678:
633:
632:
610:
607:
581:differentiable
573:is finite. If
540:
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513:
510:
507:
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501:
498:
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492:
489:
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477:
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467:
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462:lim sup
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456:
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450:
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444:
440:
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419:is defined by
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309:lim inf
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244:supremum limit
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185:
176:
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161:lim sup
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141:
137:
106:is denoted by
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92:
87:
82:
77:
72:
69:
33:Dini derivates
15:
13:
10:
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2:
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1498:
1497:Real analysis
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1432:Real Analysis
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1398:0-13-067389-7
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1389:Prentice Hall
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1240:
1234:
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1221:
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1187:
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981:
976:
972:
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959:
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954:
950:
945:
940:
939:
921:
914:
911:
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902:
899:
893:
887:
878:
875:
868:
860:
854:
848:
843:
839:
831:
830:
829:
810:
803:
797:
794:
788:
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782:
776:
767:
764:
757:
749:
743:
737:
732:
728:
720:
719:
715:
710:
694:
687:
683:
677:
670:
654:
647:
643:
640:
635:
634:
621:
617:
613:
612:
608:
606:
603:
598:
595:is the usual
593:
587:
582:
577:
559:
554:
551:
546:
526:
521:
514:
508:
505:
499:
496:
493:
490:
484:
475:
472:
465:
457:
451:
448:
445:
438:
434:
430:
422:
421:
420:
417:
411:
406:
401:
395:
393:
392:infimum limit
370:
365:
358:
355:
352:
346:
343:
337:
331:
322:
319:
312:
304:
298:
291:
287:
283:
275:
274:
273:
258:
253:
249:
245:
222:
217:
210:
204:
201:
195:
192:
189:
183:
174:
171:
164:
156:
150:
143:
139:
135:
127:
126:
125:
110:
90:
70:
67:
60:
59:
58:
57:
53:
49:
44:
42:
38:
34:
30:
26:
25:real analysis
22:
1469:
1468:
1450:
1431:
1416:
1383:
1183:
1071:
943:
827:
708:
692:
685:
681:
675:
668:
652:
645:
641:
638:
601:
591:
585:
575:
557:
544:
541:
415:
409:
405:vector space
399:
396:
385:
256:
251:
237:
108:
105:
51:
47:
45:
32:
28:
18:
41:Ulisse Dini
21:mathematics
1486:Categories
1472:PlanetMath
1366:References
597:derivative
37:derivative
1423:EMS Press
1280:−
1255:−
1156:−
1144:−
1116:→
1088:−
1039:−
1005:→
912:−
900:−
872:→
844:−
795:−
761:→
553:Lipschitz
506:−
469:→
356:−
344:−
316:→
288:−
202:−
168:→
81:→
1339:See also
1332:extended
1314: .
1188:and the
1186:supremum
953:supremum
620:supremum
439:′
292:′
144:′
1334:sense).
1318:On the
1190:infimum
949:infimum
616:infimum
609:Remarks
555:, then
550:locally
390:is the
388:lim inf
242:is the
240:lim sup
54:, of a
1457:
1438:
1395:
955:limit.
386:where
250:. The
238:where
27:, the
716:Also,
1455:ISBN
1436:ISBN
1393:ISBN
1072:and
828:and
673:and
618:and
46:The
31:(or
1326:or
951:or
599:at
583:at
579:is
548:is
542:If
397:If
19:In
1488::
1421:,
1415:,
1391:.
1373:^
1328:ââ
1324:+â
605:.
394:.
254:,
1478:.
1463:.
1444:.
1426:.
1401:.
1312:t
1308:f
1294:)
1291:t
1288:(
1285:f
1276:D
1272:=
1269:)
1266:t
1263:(
1260:f
1251:D
1247:=
1244:)
1241:t
1238:(
1235:f
1230:+
1226:D
1222:=
1219:)
1216:t
1213:(
1210:f
1205:+
1201:D
1180:.
1166:h
1162:)
1159:h
1153:t
1150:(
1147:f
1141:)
1138:t
1135:(
1132:f
1123:+
1120:0
1113:h
1105:=
1102:)
1099:t
1096:(
1093:f
1084:D
1055:h
1051:)
1048:t
1045:(
1042:f
1036:)
1033:h
1030:+
1027:t
1024:(
1021:f
1012:+
1009:0
1002:h
994:=
991:)
988:t
985:(
982:f
977:+
973:D
944:D
936:.
922:h
918:)
915:h
909:t
906:(
903:f
897:)
894:t
891:(
888:f
879:+
876:0
869:h
861:=
858:)
855:t
852:(
849:f
840:D
811:h
807:)
804:t
801:(
798:f
792:)
789:h
786:+
783:t
780:(
777:f
768:+
765:0
758:h
750:=
747:)
744:t
741:(
738:f
733:+
729:D
713:.
711:)
709:t
707:(
702:â
698:â˛
693:f
688:)
686:t
684:(
682:f
679:â
676:D
671:)
669:t
667:(
662:+
658:â˛
653:f
648:)
646:t
644:(
642:f
639:D
629:â
625:t
602:t
592:t
586:t
576:f
567:+
563:â˛
558:f
545:f
527:.
522:h
518:)
515:t
512:(
509:f
503:)
500:d
497:h
494:+
491:t
488:(
485:f
476:+
473:0
466:h
458:=
455:)
452:d
449:,
446:t
443:(
435:+
431:f
416:d
410:t
400:f
371:,
366:h
362:)
359:h
353:t
350:(
347:f
341:)
338:t
335:(
332:f
323:+
320:0
313:h
305:=
302:)
299:t
296:(
284:f
266:â
262:â˛
257:f
223:,
218:h
214:)
211:t
208:(
205:f
199:)
196:h
193:+
190:t
187:(
184:f
175:+
172:0
165:h
157:=
154:)
151:t
148:(
140:+
136:f
118:+
114:â˛
109:f
91:,
86:R
76:R
71::
68:f
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