720:
360:
584:
608:. S-continuity is referred to as an external property. The first definition is external because it involves quantification over standard values only. The second definition is external because it involves the external relation of being infinitesimal.
227:
450:
918:
465:
355:{\displaystyle \forall \epsilon \in \mathbb {R} ^{+},\exists \delta \in \mathbb {R} ^{+},|h|\leq \delta \implies |f(x+h)-f(x)|\leq \varepsilon }
923:
652:
371:
866:
759:
703:
690:
42:
by
Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard
579:{\displaystyle \forall {\mbox{ positive }}\delta \cong 0,\ (|h|\leq \delta \implies |f(x+h)-f(x)|<\varepsilon ).}
749:
949:
754:
831:
645:
455:
The proof that the second fact implies the first uses overspill, since given a non-infinitesimal positive
764:
780:
882:
709:
638:
811:
65:
185:
If an internal set contains all infinitesimal non-negative hyperreals, it contains a positive
897:
680:
739:
734:
27:
8:
744:
213:
These facts can be used to prove the equivalence of the following two conditions for an
928:
816:
73:
589:
Applying overspill, we obtain a positive appreciable δ with the requisite properties.
851:
841:
826:
892:
887:
785:
617:
20:
592:
These equivalent conditions express the property known in nonstandard analysis as
861:
846:
685:
601:
902:
795:
148:
were an internal set, then instantiating the internal induction principle with
43:
943:
661:
174:
836:
821:
58:
50:
790:
696:
31:
675:
622:
Lectures on the hyperreals. An introduction to nonstandard analysis.
719:
630:
163:
The overspill principle has a number of useful consequences:
19:
This article is about mathematics. For housing estates, see
445:{\displaystyle \forall h\cong 0,\ |f(x+h)-f(x)|\cong 0}
473:
468:
374:
230:
578:
444:
354:
941:
200:it contains an unlimited (infinite) element of *
167:The set of standard hyperreals is not internal.
170:The set of bounded hyperreals is not internal.
646:
653:
639:
520:
516:
302:
298:
266:
242:
217:hyperreal-valued function ƒ defined on *
16:Proof technique in nonstandard analysis
942:
765:Infinitesimal strain theory (physics)
634:
160:which is known not to be the case.
13:
660:
469:
375:
255:
231:
14:
961:
867:Transcendental law of homogeneity
760:Constructive nonstandard analysis
704:The Method of Mechanical Theorems
691:Criticism of nonstandard analysis
718:
750:Synthetic differential geometry
120: + 1 also belongs to
570:
560:
556:
550:
541:
529:
522:
517:
506:
498:
494:
432:
428:
422:
413:
401:
394:
342:
338:
332:
323:
311:
304:
299:
288:
280:
1:
919:Analyse des Infiniment Petits
755:Smooth infinitesimal analysis
611:
196:If an internal set contains
7:
177:hyperreals is not internal.
10:
966:
208:
68:for the standard integers
18:
911:
883:Gottfried Wilhelm Leibniz
875:
804:
773:
727:
716:
668:
76:we get the principle of
812:Standard part function
580:
446:
356:
898:Augustin-Louis Cauchy
710:Cavalieri's principle
581:
447:
357:
53:of the internal set *
950:Nonstandard analysis
740:Nonstandard calculus
735:Nonstandard analysis
475: positive
466:
372:
228:
28:nonstandard analysis
924:Elementary Calculus
805:Individual concepts
745:Internal set theory
101:1 is an element of
66:induction principle
817:Transfer principle
681:Leibniz's notation
576:
477:
442:
352:
152:, it would follow
108:for every element
78:internal induction
74:transfer principle
937:
936:
852:Law of continuity
842:Levi-Civita field
827:Increment theorem
786:Hyperreal numbers
493:
476:
392:
187:non-infinitesimal
957:
893:Pierre de Fermat
888:Abraham Robinson
728:Related branches
722:
655:
648:
641:
632:
631:
618:Robert Goldblatt
585:
583:
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563:
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361:
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307:
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275:
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269:
251:
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64:By applying the
38:(referred to as
21:overspill estate
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960:
959:
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955:
954:
940:
939:
938:
933:
929:Cours d'Analyse
907:
871:
862:Microcontinuity
847:Hyperfinite set
800:
796:Surreal numbers
769:
723:
714:
686:Integral symbol
664:
659:
628:
614:
602:microcontinuity
559:
521:
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431:
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303:
287:
279:
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229:
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211:
181:In particular:
51:internal subset
44:natural numbers
24:
17:
12:
11:
5:
963:
953:
952:
935:
934:
932:
931:
926:
921:
915:
913:
909:
908:
906:
905:
903:Leonhard Euler
900:
895:
890:
885:
879:
877:
876:Mathematicians
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859:
854:
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824:
819:
814:
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799:
798:
793:
788:
783:
777:
775:
774:Formalizations
771:
770:
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767:
762:
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752:
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742:
737:
731:
729:
725:
724:
717:
715:
713:
712:
707:
700:
693:
688:
683:
678:
672:
670:
666:
665:
662:Infinitesimals
658:
657:
650:
643:
635:
626:
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613:
610:
587:
586:
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572:
569:
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453:
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412:
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351:
348:
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331:
328:
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316:
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306:
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297:
294:
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282:
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263:
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257:
254:
249:
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236:
233:
210:
207:
206:
205:
194:
179:
178:
171:
168:
156: = *
142:
141:
137: = *
128:
127:
126:
125:
106:
30:, a branch of
15:
9:
6:
4:
3:
2:
962:
951:
948:
947:
945:
930:
927:
925:
922:
920:
917:
916:
914:
910:
904:
901:
899:
896:
894:
891:
889:
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884:
881:
880:
878:
874:
868:
865:
863:
860:
858:
855:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
809:
807:
803:
797:
794:
792:
789:
787:
784:
782:
781:Differentials
779:
778:
776:
772:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
732:
730:
726:
721:
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708:
706:
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701:
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698:
694:
692:
689:
687:
684:
682:
679:
677:
674:
673:
671:
667:
663:
656:
651:
649:
644:
642:
637:
636:
633:
629:
623:
619:
616:
615:
609:
607:
603:
599:
595:
590:
573:
567:
564:
553:
547:
544:
538:
535:
532:
526:
513:
510:
502:
488:
485:
482:
479:
462:
461:
460:
458:
439:
436:
425:
419:
416:
410:
407:
404:
398:
387:
384:
381:
378:
368:
367:
366:
349:
346:
335:
329:
326:
320:
317:
314:
308:
295:
292:
284:
276:
271:
261:
258:
252:
247:
237:
234:
224:
223:
222:
220:
216:
203:
199:
195:
192:
188:
184:
183:
182:
176:
175:infinitesimal
172:
169:
166:
165:
164:
161:
159:
155:
151:
147:
140:
136:
133:
132:
131:
123:
119:
115:
111:
107:
104:
100:
99:
98:
97:
96:
94:
90:
86:
81:
79:
75:
71:
67:
62:
60:
56:
52:
48:
45:
41:
37:
33:
29:
22:
856:
837:Internal set
822:Hyperinteger
791:Dual numbers
702:
695:
627:
621:
605:
597:
593:
591:
588:
456:
454:
364:
218:
214:
212:
201:
197:
193:) hyperreal.
190:
186:
180:
162:
157:
153:
149:
145:
143:
138:
134:
129:
121:
117:
113:
109:
102:
92:
88:
84:
82:
77:
69:
63:
59:hypernatural
54:
46:
39:
35:
25:
697:The Analyst
191:appreciable
173:The set of
32:mathematics
676:Adequality
612:References
604:) of ƒ at
598:continuity
61:numbers.
49:is not an
912:Textbooks
857:Overspill
624:Springer.
568:ε
545:−
518:⟹
514:δ
511:≤
483:≅
480:δ
470:∀
437:≅
417:−
382:≅
376:∀
350:ε
347:≤
327:−
300:⟹
296:δ
293:≤
262:∈
259:δ
256:∃
238:∈
235:ϵ
232:∀
36:overspill
944:Category
620:(1998).
215:internal
85:internal
83:For any
72:and the
40:overflow
669:History
209:Example
87:subset
492:
391:
95:, if
832:Monad
130:then
105:, and
600:(or
565:<
365:and
189:(or
91:of *
144:If
116:,
112:of
57:of
26:In
946::
459:,
221:.
80::
34:,
654:e
647:t
640:v
606:x
596:-
594:S
574:.
571:)
561:|
557:)
554:x
551:(
548:f
542:)
539:h
536:+
533:x
530:(
527:f
523:|
507:|
503:h
499:|
495:(
489:,
486:0
457:ε
440:0
433:|
429:)
426:x
423:(
420:f
414:)
411:h
408:+
405:x
402:(
399:f
395:|
388:,
385:0
379:h
343:|
339:)
336:x
333:(
330:f
324:)
321:h
318:+
315:x
312:(
309:f
305:|
289:|
285:h
281:|
277:,
272:+
267:R
253:,
248:+
243:R
219:R
204:.
202:N
198:N
158:N
154:N
150:N
146:N
139:N
135:A
124:,
122:A
118:n
114:A
110:n
103:A
93:N
89:A
70:N
55:N
47:N
23:.
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