7.12. Folgen differenzierbarer Funktionen


Wir nehmen die Überlegungen aus Abschnitt 6.10 wieder auf. Dort konnten wir zeigen, dass die gleichmäßige Konvergenz mit der Stetigkeit verträglich ist.

Eine direkte Übertragung auf differenzierbare Funktionen gelingt allerdings nicht, denn wir wissen etwa aus dem Beweis des Weierstraßschen Approximationssatzes, dass jede stetige Funktion auf [0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGymaiaac2faaaa@39D1@ gleichmäßiger Grenzwert einer Polynomfolge, also einer Folge differenzierbarer Funktionen ist (siehe [6.7.2]). So ist z.B. die (in 1 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@3773@ ) nicht differenzierbare Funktion |X 1 2 | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIfacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaiaacYhaaaa@3B3D@ auf [0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGymaiaac2faaaa@39D1@ als gleichmäßiger Limes einer Folge differenzierbarer Funktionen zu gewinnen.

Aber selbst wenn die Grenzfunktion wieder differenzierbar ist, muss ihre Ableitung nicht aus den Ableitungen der Folgenglieder hervorgehen. So weiß man etwa mit dem Majorantenkriterium [6.10.7] dass

1 n sin(nX) gm 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaamOBaaaaciGGZbGaaiyAaiaac6gacqWIyiYBcaGGOaGaamOBaiaadIfacaGGPaWaaCbeaeaacqGHsgIRaSqaaiaadEgacaWGTbaabeaakiaaicdaaaa@43AD@ ,

denn | 1 n sin(nx)| 1 n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFamaalaaabaGaaGymaaqaaiaad6gaaaGaci4CaiaacMgacaGGUbGaaiikaiaad6gacaWG4bGaaiykaiaacYhacqGHKjYOdaWcaaqaaiaaigdaaeaacaWGUbaaaaaa@433E@ für alle n und alle x, aber die Folge (( 1 n sinnX ) )=(cosnX) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcadaWcaaqaaiaaigdaaeaacaWGUbaaaiGacohacaGGPbGaaiOBaiablIHiVjaad6gacaWGybGabiykayaafaGaaiykaiabg2da9iaacIcaciGGJbGaai4BaiaacohacqWIyiYBcaWGUbGaamiwaiaacMcaaaa@4886@ ist nicht einmal punktweise konvergent: (cos(nπ))=( (1) n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiGacogacaGGVbGaai4CaiaacIcacaWGUbGaeqiWdaNaaiykaiaacMcacqGH9aqpcaGGOaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaaiykaaaa@44AB@ z.B. ist divergent.

Interessanterweise läßt sich die Differenzierbarkeit der Limesfunktion mit passender Ableitung gewinnen, wenn die gleichmäßige Konvergenz der Ableitungsfolge gegeben ist. Für die Funktionenfolge selbst muss lediglich die punktweise Konvergenz gesichert sein.

Der Nachweis verwendet, wieder einmal, dem Mittelwertsatz, so dass die Gültigkeit an Intervalle gebunden ist, in unserem Fall sogar an beschränkte Intervalle.

Bemerkung:  Sei I ein beschränktes Intervall,  f n D 1 (I) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakiabgIGiolaadseadaahaaWcbeqaaiaaigdaaaGccaGGOaGaamysaiaacMcaaaa@3D66@ und g:I MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaacQdacaWGjbGaeyOKH4QaeSyhHekaaa@3BC1@ , so dass f n gm g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaWaaSbaaSqaaiaad6gaaeqaaOWaaCbeaeaacqGHsgIRaSqaaiaadEgacaWGTbaabeaakiaadEgaaaa@3D06@ . Ist für ein aI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiolaadMeaaaa@3924@ die Zahlenfolge ( f n (a)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyyaiaacMcacaGGPaaaaa@3B98@ konvergent, so gilt der Reihe nach:

  1. ( f n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3959@ ist gleichmäßig konvergent

[7.12.1]
  1. flim f n :I MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iGacYgacaGGPbGaaiyBaiaadAgadaWgaaWcbaGaamOBaaqabaGccaGG6aGaamysaiabgkziUkabl2riHcaa@41AA@ ist differenzierbar in a und f (a)=g(a) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaaiikaiaadggacaGGPaGaeyypa0Jaam4zaiaacIcacaWGHbGaaiykaaaa@3D53@

[7.12.2]

Beweis:  

1.  Wir setzen den zentralen Darstellungssatz [7.5.1] ein und finden zu jedem n eine in a stetige Funktion r n :I MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaWGUbaabeaakiaacQdacaWGjbGaeyOKH4QaeSyhHekaaa@3CF5@ mit r n (a)= f n (a) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaWGUbaabeaakiaacIcacaWGHbGaaiykaiabg2da9iqadAgagaqbamaaBaaaleaacaWGUbaabeaakiaacIcacaWGHbGaaiykaaaa@3FB0@ , so dass

f n = f n (a)+(Xa) r n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakiabg2da9iaadAgadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyyaiaacMcacqGHRaWkcaGGOaGaamiwaiabgkHiTiaadggacaGGPaGaamOCamaaBaaaleaacaWGUbaabeaaaaa@445A@ [0]

Die so gewonnene Funktionenfolge ( r n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadkhadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3965@ erweist sich gemäß Cauchy-Kriterium [6.10.8] als gleichmäßig konvergent auf I. Ist nämlich ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3955@ beliebig vorgegeben, so gibt es nach Voraussetzung über ( f n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiqadAgagaqbamaaBaaaleaacaWGUbaabeaakiaacMcaaaa@3965@ (wieder mit dem Cauchy-Kriterium) ein n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaakiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3BDB@ , so dass

| f n (x) f m (x)|<ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiqadAgagaqbamaaBaaaleaacaWGUbaabeaakiaacIcacaWG4bGaaiykaiabgkHiTiqadAgagaqbamaaBaaaleaacaWGTbaabeaakiaacIcacaWG4bGaaiykaiaacYhacqGH8aapcqaH1oqzaaa@446F@ [1]

für alle n,m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaaaaa@3C20@ und alle xI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadMeaaaa@393B@ . Insbesondere also hat man für diese n, m:

| r n (a) r m (a)|=| f n (a) f m (a)|<ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadkhadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyyaiaacMcacqGHsislcaWGYbWaaSbaaSqaaiaad2gaaeqaaOGaaiikaiaadggacaGGPaGaaiiFaiabg2da9iaacYhaceWGMbGbauaadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyyaiaacMcacqGHsislceWGMbGbauaadaWgaaWcbaGaamyBaaqabaGccaGGOaGaamyyaiaacMcacaGG8bGaeyipaWJaeqyTdugaaa@50F1@ .

Für ein von a verschiedenes xI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadMeaaaa@393B@ gibt es nun zu n,m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaaaaa@3C20@ nach Mittelwertsatz [7.9.5] ein x ˜ nm MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaiaWaaSbaaSqaaiaad6gacaaMc8UaamyBaaqabaaaaa@3A94@ zwischen a und x, mit

( f n f m )(x)( f n f m )(a) xa =( f n f m ) ( x ˜ nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaGGOaGaamOzamaaBaaaleaacaWGUbaabeaakiabgkHiTiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGPaGaaiikaiaadIhacaGGPaGaeyOeI0IaaiikaiaadAgadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGMbWaaSbaaSqaaiaad2gaaeqaaOGaaiykaiaacIcacaWGHbGaaiykaaqaaiaadIhacqGHsislcaWGHbaaaiabg2da9iaacIcacaWGMbWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamOzamaaBaaaleaacaWGTbaabeaakiqacMcagaqbaiaacIcaceWG4bGbaGaadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@58B2@ ,

so dass mit [1] die folgende Abschätzung gelingt:

| r n (x) r m (x)| =| f n (x) f n (a) xa f m (x) f m (a) xa | =| ( f n f m )(x)( f n f m )(a) xa | =|( f n f m ) ( x ˜ nm )| =| f n ( x ˜ nm ) f m ( x ˜ nm )|<ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9D9D@

Mit der gleichmäßigen Konvergenz von ( r n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadkhadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3965@ zeigen wir jetzt die eigentliche Behauptung. Dazu geben wir wieder ein ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3955@ vor. Da I beschränkt ist, hat I einen endlichen Durchmesser c>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg6da+iaaicdaaaa@3896@ . Zu ε 2c >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaH1oqzaeaacaaIYaGaam4yaaaacqGH+aGpcaaIWaaaaa@3B09@ gibt es nun ein n 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3BDC@ , so dass

| r n (x) r m (x)|< ε 2c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadkhadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiEaiaacMcacqGHsislcaWGYbWaaSbaaSqaaiaad2gaaeqaaOGaaiikaiaadIhacaGGPaGaaiiFaiabgYda8maalaaabaGaeqyTdugabaGaaGOmaiaadogaaaaaaa@4623@  für alle n,m n 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyyzImRaamOBamaaBaaaleaacaaIXaaabeaaaaa@3C21@ und alle xI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadMeaaaa@393B@ .[2]

Ferner finden wir ein n 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIYaaabeaakiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3BDD@ (nach [5.5.7] ist ( f n (a)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyyaiaacMcacaGGPaaaaa@3B98@ eine Cauchy-Folge) derart, dass

| f n (a) f m (a)|< ε 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyyaiaacMcacqGHsislcaWGMbWaaSbaaSqaaiaad2gaaeqaaOGaaiikaiaadggacaGGPaGaaiiFaiabgYda8maalaaabaGaeqyTdugabaGaaGOmaaaaaaa@44F5@  für alle n,m n 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyyzImRaamOBamaaBaaaleaacaaIYaaabeaaaaa@3C22@ .[3]

Für alle n,m n 0 max{ n 1 , n 2 } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaakiabg2da9iGac2gacaGGHbGaaiiEaiaacUhacaWGUbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaad6gadaWgaaWcbaGaaGOmaaqabaGccaGG9baaaa@467D@ und alle xI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadMeaaaa@393B@ folgt aus der Darstellung [0] jetzt mit [2] und [3]:

| f n (x) f m (x)| =| f n (a)+(xa) r n (x) f m (a)(xa) r m (x)| | f n (a) f m (a)|+|xa|| r n (x) r m (x)| < ε 2 +c ε 2c =ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9010@

2.  Zunächst ist der nach 1 existierende Limes rlim r n :I MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2da9iGacYgacaGGPbGaaiyBaiaadkhadaWgaaWcbaGaamOBaaqabaGccaGG6aGaamysaiabgkziUkabl2riHcaa@41C2@ gemäß [6.10.5] stetig in a. Ferner folgt für die Funktion f=lim f n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iGacYgacaGGPbGaaiyBaiaadAgadaWgaaWcbaGaamOBaaqabaaaaa@3CB7@ aus den punktweisen Konvergenzen

  • f n (a)+(Xa) r n pw f(a)+(Xa)r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakiaacIcacaWGHbGaaiykaiabgUcaRiaacIcacaWGybGaeyOeI0IaamyyaiaacMcacaWGYbWaaSbaaSqaaiaad6gaaeqaaOWaaCbeaeaacqGHsgIRaSqaaiaadchacaWG3baabeaakiaadAgacaGGOaGaamyyaiaacMcacqGHRaWkcaGGOaGaamiwaiabgkHiTiaadggacaGGPaGaamOCaaaa@4E77@

  • f n (a)+(Xa) r n = f n pw f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakiaacIcacaWGHbGaaiykaiabgUcaRiaacIcacaWGybGaeyOeI0IaamyyaiaacMcacaWGYbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaamOzamaaBaaaleaacaWGUbaabeaakmaaxababaGaeyOKH4kaleaacaWGWbGaam4DaaqabaGccaWGMbaaaa@4970@

die Identität f=f(a)+(Xa)r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaadAgacaGGOaGaamyyaiaacMcacqGHRaWkcaGGOaGaamiwaiabgkHiTiaadggacaGGPaGaamOCaaaa@40E9@ . Dies sichert nach [7.5.1] die Differenzierbarkeit von f in a sowie die Ableitung

f (a)=r(a)=lim r n (a)=lim f n (a)=g(a) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaaiikaiaadggacaGGPaGaeyypa0JaamOCaiaacIcacaWGHbGaaiykaiabg2da9iGacYgacaGGPbGaaiyBaiaadkhadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyyaiaacMcacqGH9aqpciGGSbGaaiyAaiaac2gaceWGMbGbauaadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyyaiaacMcacqGH9aqpcaWGNbGaaiikaiaadggacaGGPaaaaa@51F9@

Das lokale Ergebnis [7.12.2] läßt sich nun globalisieren, wobei man sogar auf die Beschränktheit des Intervalls verzichten darf, und außerdem auf C 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaaGymaaaaaaa@379C@ -Funktionen übertragen.

Bemerkung:  Es sei I ein beliebiges Intervall,  f n D 1 (I) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakiabgIGiolaadseadaahaaWcbeqaaiaaigdaaaGccaGGOaGaamysaiaacMcaaaa@3D66@ , g:I MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaacQdacaWGjbGaeyOKH4QaeSyhHekaaa@3BC1@ , so dass f n |J gm g|J MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaWaaSbaaSqaaiaad6gaaeqaaOGaaiiFaiaadQeadaWfqaqaaiabgkziUcWcbaGaam4zaiaad2gaaeqaaOGaam4zaiaacYhacaWGkbaaaa@40A4@ für jedes abgeschlossene Teilintervall JI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabgkOimlaadMeaaaa@3985@ . Ist nun für ein aI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiolaadMeaaaa@3924@ die Zahlenfolge ( f n (a)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamyyaiaacMcacaGGPaaaaa@3B98@ konvergent, so gilt für den punktweisen Limes flim f n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iGacYgacaGGPbGaaiyBaiaadAgadaWgaaWcbaGaamOBaaqabaaaaa@3CB7@ :

  1. f D 1 (I) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadseadaahaaWcbeqaaiaaigdaaaGccaGGOaGaamysaiaacMcaaaa@3C3D@ und f =g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaeyypa0Jaam4zaaaa@38D5@

[7.12.3]
  1. f n C 1 (I)f C 1 (I) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakiabgIGiolaadoeadaahaaWcbeqaaiaaigdaaaGccaGGOaGaamysaiaacMcacaaMf8UaeyO0H4TaaGzbVlaadAgacqGHiiIZcaWGdbWaaWbaaSqabeaacaaIXaaaaOGaaiikaiaadMeacaGGPaaaaa@492E@

[7.12.4]

Beweis:  Wir zeigen zunächst, dass die Folge ( f n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3959@ punktweise konvergiert. Sei dazu xI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadMeaaaa@393B@ beliebig, o.E. xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgcMi5kaadggaaaa@3996@ , x und a spannen ein abgeschlossenes Teilintervall JI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabgkOimlaadMeaaaa@3985@ auf, so dass nach Voraussetzung ( f n |J) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiqadAgagaqbamaaBaaaleaacaWGUbaabeaakiaacYhacaWGkbGaaiykaaaa@3B34@ gleichmäßig konvergiert. Mit [7.12.1] ist damit ( f n |J) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamOBaaqabaGccaGG8bGaamOsaiaacMcaaaa@3B28@ ebenfalls gleichmäßig, also auch punktweise konvergent. Da xJ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadQeaaaa@393C@ , konvergiert insbesondere die Zahlenfolge ( f n (x)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiEaiaacMcacaGGPaaaaa@3BAF@ .

1.   Gemäß Vorüberlegung konvergiert ( f n (x)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiEaiaacMcacaGGPaaaaa@3BAF@ für jedes xI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadMeaaaa@393B@ . Da es zu jedem x eine relative ε-Umgebung  I x,ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWG4bGaaiilaiabew7aLbqabaaaaa@3A3A@

 i

ε 1 2 { min{|xa|,|xb| ,1 ) },  falls x kein Randpunkt von I ist min{|ba| ,1 ) },  falls x ein Randpunkt von I ist MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9659@

ist eine nicht-negative Zahl, die mit

J{ [xε,x+ε] ,  falls x kein Randpunkt von I ist [x,x+ε] ,  falls  x=a [xε,x] ,  falls  x=b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@80A6@

ein abgeschlossenes Teilintervall von I liefert, so dass I x,ε =I]xε,x+ε[J MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWG4bGaaiilaiabew7aLbqabaGccqGH9aqpcaWGjbGaeyykICSaaiyxaiaadIhacqGHsislcqaH1oqzcaGGSaGaamiEaiabgUcaRiabew7aLjaacUfacqGHckcZcaWGkbaaaa@4A08@ .

__________
*) Mit der Vereinbarung |x±|=|+|=>r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHXcqScqGHEisPcaGG8bGaeyypa0JaaiiFaiabg6HiLkabgUcaRiabg6HiLkaacYhacqGH9aqpcqGHEisPcqGH+aGpcaWGYbaaaa@4788@ für jedes reelle r berücksichtigen wir so den Fall unbeschränkter Intervalle.

gibt, die in einem abgeschlossenen Teilintervall J von I liegt, garantiert [7.12.2] die Differenzierbarkeit von f| I x,ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacYhacaWGjbWaaSbaaSqaaiaadIhacaGGSaGaeqyTdugabeaaaaa@3C25@ in x mit (f| I x,ε ) (x)=g(x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgacaGG8bGaamysamaaBaaaleaacaWG4bGaaiilaiabew7aLbqabaGcceGGPaGbauaacaGGOaGaamiEaiaacMcacqGH9aqpcaWGNbGaaiikaiaadIhacaGGPaaaaa@4432@ . Auf Grund des lokalen Charakters der Differenzierbarkeit (siehe [7.4.2]) ist dies die Behauptung.

2.   Sind alle f n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaWaaSbaaSqaaiaad6gaaeqaaaaa@3802@ stetig, so ist nach [6.10.6] auch f =g=lim f n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaeyypa0Jaam4zaiabg2da9iGacYgacaGGPbGaaiyBaiqadAgagaqbamaaBaaaleaacaWGUbaabeaaaaa@3EC1@ als lokal gleichmäßiger Limes der Folge ( f n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiqadAgagaqbamaaBaaaleaacaWGUbaabeaakiaacMcaaaa@3965@ stetig.


7.11.