Home Analysis • Download e-book for kindle: Analysis 2: Differentialrechnung im IRn, gewöhnliche by Otto Forster

Download e-book for kindle: Analysis 2: Differentialrechnung im IRn, gewöhnliche by Otto Forster

By Otto Forster

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Extra resources for Analysis 2: Differentialrechnung im IRn, gewöhnliche Differentialgleichungen, 9. Auflage

Example text

Nach dem Satz über die Approximation von Integralen durch Riemannsche Summen (An. 1, § 18, Satz 8) existiert ein 1)1 > 0, so dass la I b 11/(t)lldt- ~ 11/(ti)ll(ti-ti-l)1 ::;; ~ (1) für jede Unterteilung a = to < t1 < ... < tk = b (*) des Intervalls [a,b] der Feinheit::;; 1)1. d. 7) Sei

O. Wir betrachten den Kreisbogen I : [O,

1, § 18, Satz 8) existiert ein 1)1 > 0, so dass la I b 11/(t)lldt- ~ 11/(ti)ll(ti-ti-l)1 ::;; ~ (1) für jede Unterteilung a = to < t1 < ... < tk = b (*) des Intervalls [a,b] der Feinheit::;; 1)1. d. 7) Sei

O. Wir betrachten den Kreisbogen I : [O,

I), k~1. Es gilt r(ak) = Vii/k, also F(a ) = (l/k)n =n-n/2 (unabhängigvonk). k (Vii/k)n Da limk->=F(ak) = n-n/2 undF(O) = 0, istF im Nullpunkt nicht stetig. I. Differentialrechnung im )Rn 56 Für Funktionen mehrerer Veränderlichen folgt also, im Gegensatz zum Fall n = 1, aus der partiellen Differenzierbarkeit nicht die Stetigkeit. Bemerkung. Wir werden im nächsten Paragraphen eine Verallgemeinerung des Differenzierbarkeitsbegriffs auf Funktionen mehrerer Veränderlichen kennen lernen, welche die Stetigkeit nach sich zieht.

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