84:
74:
53:
22:
1244:
The statement of the theorem is that there exists a c in the open interval (a,b). Yet the very first line of the proof states that one can choose c to be either a or b. This is wrong. One must either relax the condition and let c be in the closed interval or prove the existence of such a c in the
1217:
Why does nobody write what the implications of this theorem are? In the Intro it says "But even when ƒ′ is not continuous, Darboux's theorem places a severe restriction on what it can be.". Exactly that is the question: What severe restrictions? This is one of the most important parts of the theorem
918:
What can be said in the context of your proof of
Darboux theorem is that in a right neighbourhood of a there exists a y such that f(a)<f(y)(since as the function was differentiable the limit f(a)- f(x)/x-a existed thus was positive in the right neighbourhood) thus the max is to the right of a and
870:
The article currently says "Another proof based solely on the mean value theorem and the intermediate value theorem is due to Lars Olsen," and cites Olsen, Lars: A New Proof of
Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical Monthly. The proof being referred
227:
What can be said in the context of your proof of
Darboux theorem is that in a right neighbourhood of a there exists a y such that f(a)<f(y)(since as the function was differentiable the limit f(a)- f(x)/x-a existed thus was positive in the right neighbourhood) thus the max is to the right of a and
915:
Hi, just a small point. Since derivatives are not necessarily continuous we can't say that functions in general are decreasing or increasing in a neighbourhood of a point regardless of the value of the derivative there. You can create examples easily enough by considering functions that oscillate
223:
Hi, just a small point. Since derivatives are not necessarily continuous we can't say that functions in general are decreasing or increasing in a neighbourhood of a point regardless of the value of the derivative there. You can create examples easily enough by considering functions that oscillate
923:
just to labour the point saying the function increases in a right-neighbourhood of a says that if x, y belong to that neighbourhood and x lies between a and y then f(x)<f(y). However all that is implied by the derivative taking a positive value at a is that f(a)<f(x) and f(a)<f(y).
231:
just to labour the point saying the function increases in a right-neighbourhood of a says that if x, y belong to that neighbourhood and x lies between a and y then f(x)<f(y). However all that is implied by the derivative taking a positive value at a is that f(a)<f(x) and f(a)<f(y).
1190:
1080:
871:
to is not due to Lars Olsen. Despite the title "A New Proof of
Darboux's Theorem," Olsen's proof is not new. The proof appears, for example, in Mathematical Analysis (2e) by Tom M. Apostol. See Theorem 5.16 of that book.
946:
First usage of Ferma's theorem is wrong because you can use it on (a, b) and not . for example for function f(x) = x^2 on f has maximums on -2 and 2 (f(2) = f(-2) = 4) and derivation of f on neither -2 nor 2 is 0.
1201:
641:
140:
1376:
After the sentence beginning "In particular, the derivative of the function" should come the sentence "This function is used in the construction of
Volterra's function.", with a link.
533:
1350:
To define the IVP, I think, is necessary to have a connected domain. In the cited work (Ciesielski) is for instance the real line. Otherwise we have also or a generic interval I.
1092:
985:
739:
320:
699:
819:
673:
565:
417:
1325:
1305:
1282:
489:
469:
392:
340:
449:
372:
842:
The article says the proof based on Fermat's theorem and its corollary is due to Lars Olsen. But I have read exactly the same proof from much older text.
1404:
130:
1399:
106:
1252:
930:
242:
1225:
1357:
954:
1205:
1264:
Nope. The first sentence of the proof shows the property of the function, which is then used to apply the extreme value theorem for
97:
58:
901:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
858:
185:
170:
261:
The proof of the theorem mention's Fermat's theorem but I don't think it is quite Fermat's theorem although it is related.
33:
180:
No, it was correct. The problem was the derivative symbol that was not clearly visible. I have added a thin space. --
570:
876:
827:
181:
166:
1256:
269:
282:
You are absolutely correct. The first mention of Fermat's theorem was not correct. I removed it. It's true that
1229:
1185:{\displaystyle x\mapsto {\begin{cases}\sin(1/x)&{\text{for }}x\neq 0,\\0&{\text{for }}x=0.\end{cases}}}
934:
246:
21:
1075:{\displaystyle x\mapsto {\begin{cases}\sin(1/x)&{\text{for }}x\neq 0\\0&{\text{for }}x=0\end{cases}}}
1361:
958:
872:
823:
1381:
494:
263:
39:
1377:
1086:
Note the location of the period at the end of the sentence in the display above. I changed it to this:
83:
276:
1353:
1248:
1221:
1197:
950:
916:'uncontrollably' out of respect of my lecturer I won't give out his examples here but they do exist.
846:
224:'uncontrollably' out of respect of my lecturer I won't give out his examples here but they do exist.
1107:
1000:
744:
704:
285:
854:
678:
646:
538:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
89:
73:
52:
1385:
1365:
1336:
1332:
1233:
962:
938:
880:
862:
831:
250:
213:
201:
189:
174:
163:
402:
209:
162:
The statement of the theorem is wrong. Since the very beginning. In the original source,
850:
1310:
1290:
1267:
474:
454:
377:
325:
1393:
898:
422:
345:
971:
Last edited at 17:01, 3 November 2012 (UTC). Substituted at 01:58, 5 May 2016 (UTC)
1328:
102:
205:
79:
979:
An amazingly extreme case of inattentiveness to what one is doing was this:
204:
should be merged with this one since the content has significant overlap.
927:
anyway hope i used the discussion feature correctly hear goes
322:, so Fermat's theorem tells us a local maximum cannot occur at
15:
219:
In Proof: functions are not in general increasing/decreasing.
1178:
1068:
342:. But we are trying to argue that the absolute maximum on
1346:
Definition of intermediate value property of a function.
893:
1313:
1293:
1270:
1095:
988:
748:
707:
681:
649:
573:
541:
497:
477:
457:
425:
405:
380:
348:
328:
288:
101:, a collaborative effort to improve the coverage of
1319:
1299:
1276:
1184:
1074:
812:
733:
693:
667:
635:
559:
527:
483:
463:
443:
411:
386:
366:
334:
314:
897:, and are posted here for posterity. Following
636:{\displaystyle (\phi (a)-\phi (t))/(a-t)\leq 0}
944:== Wrong implementation of Ferma's theorem ==
891:The comment(s) below were originally left at
238:Hope this is in the right section this time
8:
399:Here is the reason the absolute maximum of
19:
1351:
1246:
1219:
1195:
903:
894:Talk:Darboux's theorem (analysis)/Comments
47:
1312:
1292:
1269:
1161:
1135:
1122:
1102:
1094:
1051:
1028:
1015:
995:
987:
747:
706:
680:
648:
607:
572:
540:
496:
476:
456:
424:
404:
379:
347:
327:
287:
471:. If the absolute maximum did occur at
49:
1202:2601:445:437F:FE66:900:C28F:955A:E198
813:{\displaystyle \phi '(a)=f'(a)-y: -->
528:{\displaystyle \phi (a)\geq \phi (t)}
7:
1327:. It does not extend the theorem. --
1218:and nobody seems to talk about it?
95:This article is within the scope of
38:It is of interest to the following
14:
1405:Mid-priority mathematics articles
899:several discussions in past years
115:Knowledge:WikiProject Mathematics
1400:Start-Class mathematics articles
118:Template:WikiProject Mathematics
82:
72:
51:
20:
734:{\displaystyle \phi '(a)\leq 0}
315:{\displaystyle \phi '(a)\neq 0}
135:This article has been rated as
1130:
1116:
1099:
1023:
1009:
992:
783:
777:
763:
757:
722:
716:
694:{\displaystyle t\rightarrow a}
685:
624:
612:
604:
601:
595:
586:
580:
574:
522:
516:
507:
501:
438:
426:
361:
349:
303:
297:
1:
1240:Incorrect Proof at beginning.
881:13:33, 4 September 2016 (UTC)
863:13:21, 24 November 2015 (UTC)
832:15:10, 4 September 2016 (UTC)
668:{\displaystyle a\leq t\leq b}
560:{\displaystyle a\leq t\leq b}
175:01:36, 11 February 2008 (UTC)
158:Wrong from the very beginning
109:and see a list of open tasks.
1337:16:37, 24 October 2018 (UTC)
1234:18:01, 21 January 2018 (UTC)
963:17:00, 3 November 2012 (UTC)
939:16:16, 28 October 2009 (UTC)
741:. But this contradicts that
251:16:28, 28 October 2009 (UTC)
1421:
1372:missing reference and link
214:08:53, 21 April 2009 (UTC)
1386:06:51, 3 March 2024 (UTC)
1245:open interval correctly.
914:
277:02:06, 23 July 2013 (UTC)
134:
67:
46:
1366:17:42, 29 May 2019 (UTC)
1259:) 14:32, 24 October 2018
141:project's priority scale
256:
190:10:12, 7 May 2008 (UTC)
98:WikiProject Mathematics
1321:
1301:
1278:
1186:
1076:
815:
735:
695:
669:
637:
561:
529:
485:
465:
445:
413:
388:
368:
336:
316:
28:This article is rated
1322:
1302:
1279:
1187:
1077:
975:imbecilic punctuation
816:
736:
696:
670:
638:
562:
530:
486:
466:
446:
414:
412:{\displaystyle \phi }
389:
369:
337:
317:
182:Fioravante Patrone en
167:Fioravante Patrone en
1311:
1291:
1268:
1093:
986:
746:
705:
679:
647:
571:
539:
495:
475:
455:
423:
403:
378:
346:
326:
286:
121:mathematics articles
165:, it is correct. --
1317:
1297:
1274:
1182:
1177:
1072:
1067:
887:Assessment comment
810:
731:
691:
665:
633:
557:
525:
481:
461:
441:
409:
384:
364:
332:
312:
90:Mathematics portal
34:content assessment
1368:
1356:comment added by
1320:{\displaystyle b}
1300:{\displaystyle a}
1277:{\displaystyle c}
1260:
1251:comment added by
1236:
1224:comment added by
1209:
1200:comment added by
1164:
1138:
1054:
1031:
969:
968:
953:comment added by
866:
849:comment added by
484:{\displaystyle a}
464:{\displaystyle a}
387:{\displaystyle a}
335:{\displaystyle a}
155:
154:
151:
150:
147:
146:
1412:
1326:
1324:
1323:
1318:
1306:
1304:
1303:
1298:
1283:
1281:
1280:
1275:
1208:) 14 August 2017
1191:
1189:
1188:
1183:
1181:
1180:
1165:
1162:
1139:
1136:
1126:
1081:
1079:
1078:
1073:
1071:
1070:
1055:
1052:
1032:
1029:
1019:
965:
910:Press to view →
904:
896:
873:LoonetteTheClown
865:
843:
824:LoonetteTheClown
821:
818:
817:
811:
776:
756:
740:
738:
737:
732:
715:
700:
698:
697:
692:
674:
672:
671:
666:
642:
640:
639:
634:
611:
566:
564:
563:
558:
534:
532:
531:
526:
490:
488:
487:
482:
470:
468:
467:
462:
451:cannot occur at
450:
448:
447:
444:{\displaystyle }
442:
418:
416:
415:
410:
393:
391:
390:
385:
374:cannot occur at
373:
371:
370:
367:{\displaystyle }
365:
341:
339:
338:
333:
321:
319:
318:
313:
296:
272:
266:
257:Fermat's Theorem
202:Darboux function
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1420:
1419:
1415:
1414:
1413:
1411:
1410:
1409:
1390:
1389:
1374:
1348:
1309:
1308:
1289:
1288:
1266:
1265:
1253:128.163.239.158
1242:
1215:
1176:
1175:
1159:
1153:
1152:
1133:
1103:
1091:
1090:
1066:
1065:
1049:
1043:
1042:
1026:
996:
984:
983:
977:
948:
931:122.110.191.137
919:similar for b.
892:
889:
844:
840:
769:
749:
743:
742:
708:
703:
702:
677:
676:
645:
644:
569:
568:
567:. This implies
537:
536:
493:
492:
473:
472:
453:
452:
421:
420:
401:
400:
376:
375:
344:
343:
324:
323:
289:
284:
283:
270:
265:Pratyush Sarkar
264:
259:
243:122.110.191.137
228:similar for b.
221:
198:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
1418:
1416:
1408:
1407:
1402:
1392:
1391:
1373:
1370:
1347:
1344:
1342:
1340:
1339:
1316:
1296:
1273:
1241:
1238:
1226:213.55.184.241
1214:
1211:
1193:
1192:
1179:
1174:
1171:
1168:
1160:
1158:
1155:
1154:
1151:
1148:
1145:
1142:
1134:
1132:
1129:
1125:
1121:
1118:
1115:
1112:
1109:
1108:
1106:
1101:
1098:
1084:
1083:
1069:
1064:
1061:
1058:
1050:
1048:
1045:
1044:
1041:
1038:
1035:
1027:
1025:
1022:
1018:
1014:
1011:
1008:
1005:
1002:
1001:
999:
994:
991:
976:
973:
967:
966:
922:
912:
911:
908:
888:
885:
884:
883:
839:
836:
835:
834:
820:y-y=0}" /: -->
809:
806:
803:
800:
797:
794:
791:
788:
785:
782:
779:
775:
772:
768:
765:
762:
759:
755:
752:
730:
727:
724:
721:
718:
714:
711:
690:
687:
684:
675:, and letting
664:
661:
658:
655:
652:
632:
629:
626:
623:
620:
617:
614:
610:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
556:
553:
550:
547:
544:
524:
521:
518:
515:
512:
509:
506:
503:
500:
480:
460:
440:
437:
434:
431:
428:
408:
396:
395:
383:
363:
360:
357:
354:
351:
331:
311:
308:
305:
302:
299:
295:
292:
258:
255:
235:
220:
217:
197:
194:
193:
192:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1417:
1406:
1403:
1401:
1398:
1397:
1395:
1388:
1387:
1383:
1379:
1371:
1369:
1367:
1363:
1359:
1355:
1345:
1343:
1338:
1334:
1330:
1314:
1294:
1287:
1271:
1263:
1262:
1261:
1258:
1254:
1250:
1239:
1237:
1235:
1231:
1227:
1223:
1212:
1210:
1207:
1203:
1199:
1172:
1169:
1166:
1156:
1149:
1146:
1143:
1140:
1127:
1123:
1119:
1113:
1110:
1104:
1096:
1089:
1088:
1087:
1062:
1059:
1056:
1046:
1039:
1036:
1033:
1020:
1016:
1012:
1006:
1003:
997:
989:
982:
981:
980:
974:
972:
964:
960:
956:
952:
945:
942:
940:
936:
932:
928:
925:
920:
913:
909:
906:
905:
902:
900:
895:
886:
882:
878:
874:
869:
868:
867:
864:
860:
856:
852:
848:
837:
833:
829:
825:
807:
804:
801:
798:
795:
792:
789:
786:
780:
773:
770:
766:
760:
753:
750:
728:
725:
719:
712:
709:
688:
682:
662:
659:
656:
653:
650:
630:
627:
621:
618:
615:
608:
598:
592:
589:
583:
577:
554:
551:
548:
545:
542:
519:
513:
510:
504:
498:
478:
458:
435:
432:
429:
406:
398:
397:
381:
358:
355:
352:
329:
309:
306:
300:
293:
290:
281:
280:
279:
278:
275:
273:
267:
254:
252:
248:
244:
239:
236:
233:
229:
225:
218:
216:
215:
211:
207:
203:
196:Propose merge
195:
191:
187:
183:
179:
178:
177:
176:
172:
168:
164:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1378:Kontribuanto
1375:
1358:82.136.67.75
1352:— Preceding
1349:
1341:
1285:
1247:— Preceding
1243:
1220:— Preceding
1216:
1213:Implications
1196:— Preceding
1194:
1085:
978:
970:
955:2.178.30.192
949:— Preceding
943:
929:
926:
921:
917:
890:
845:— Preceding
841:
745:y-y=0}": -->
262:
260:
240:
237:
234:
230:
226:
222:
200:The article
199:
161:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1394:Categories
907:Comment(s)
838:Lars Olsen
851:Mscdancer
1354:unsigned
1249:unsigned
1222:unsigned
1198:unsigned
951:unsigned
859:contribs
847:unsigned
643:for all
535:for all
241:regards
1286:between
491:, then
139:on the
1329:CiaPan
1284:being
814:y-y=0}
701:gives
36:scale.
793:: -->
206:McKay
1382:talk
1362:talk
1333:talk
1307:and
1257:talk
1230:talk
1206:talk
1163:for
1137:for
1053:for
1030:for
959:talk
941:CWC
935:talk
877:talk
855:talk
828:talk
271:talk
253:CWC
247:talk
210:talk
186:talk
171:talk
1111:sin
1004:sin
419:on
131:Mid
1396::
1384:)
1364:)
1335:)
1232:)
1173:0.
1144:≠
1114:
1100:↦
1037:≠
1007:
993:↦
961:)
937:)
879:)
861:)
857:•
830:)
822:.
799:−
787:−
751:ϕ
726:≤
710:ϕ
686:→
660:≤
654:≤
628:≤
619:−
593:ϕ
590:−
578:ϕ
552:≤
546:≤
514:ϕ
511:≥
499:ϕ
407:ϕ
307:≠
291:ϕ
249:)
212:)
188:)
173:)
1380:(
1360:(
1331:(
1315:b
1295:a
1272:c
1255:(
1228:(
1204:(
1170:=
1167:x
1157:0
1150:,
1147:0
1141:x
1131:)
1128:x
1124:/
1120:1
1117:(
1105:{
1097:x
1082:.
1063:0
1060:=
1057:x
1047:0
1040:0
1034:x
1024:)
1021:x
1017:/
1013:1
1010:(
998:{
990:x
957:(
933:(
875:(
853:(
826:(
808:0
805:=
802:y
796:y
790:y
784:)
781:a
778:(
774:′
771:f
767:=
764:)
761:a
758:(
754:′
729:0
723:)
720:a
717:(
713:′
689:a
683:t
663:b
657:t
651:a
631:0
625:)
622:t
616:a
613:(
609:/
605:)
602:)
599:t
596:(
587:)
584:a
581:(
575:(
555:b
549:t
543:a
523:)
520:t
517:(
508:)
505:a
502:(
479:a
459:a
439:]
436:b
433:,
430:a
427:[
394:.
382:a
362:]
359:b
356:,
353:a
350:[
330:a
310:0
304:)
301:a
298:(
294:′
274:)
268:(
245:(
208:(
184:(
169:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
