Analysis I - IV by Hilgert J.

By Hilgert J.

Der vorliegende textual content ist eine vorlaufige Ausarbeitung meiner Vorlesungen research I-IV (Wintersemester 2004/2005 { Sommersemester 2006) an der Universitat Paderborn.

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Analysis I - IV

Der vorliegende textual content ist eine vorlaufige Ausarbeitung meiner Vorlesungen research I-IV (Wintersemester 2004/2005 { Sommersemester 2006) an der Universitat Paderborn.

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Dann gibt es y1 , y2 ∈ f (I) mit f (x1 ) = y1 < y0 < y2 = f (x2 ), also x0 ∈]x1 , x2 [⊆ I. Wir k¨onnen also auch x1 , x2 ∈ I mit x0 − < x1 < x0 < x2 < x0 + finden. F¨ ur y1 := f (x1 ) und y2 := f (x2 ) gilt dann y0 ∈]y1 , y2 [ und wir finden ein δ > 0 mit y1 < y0 − δ < y0 < y0 + δ < y2 . Wenn also |y − y0 | < δ, dann gilt y1 < y < y2 und somit − + f −1 (y0 ) = x0 − < x1 = f −1 (y1 ) < f −1 (y) < f −1 (y2 ) = x2 < x0 + = f −1 (y0 ) + , was wiederum |f −1 (y) − f −1 (y0 )| < impliziert. Damit folgt die Stetigkeit von f −1 in allen Punkten von f (I) außer den Randpunkten.

2 ❅ ❅❝ ✲ 2❅ ❅ (ii) Die Funktion f : R \ {0} → R, x→ 1 1 − x x hat weder einen rechtsseitigen noch einen linksseitigen Grenzwert in 0. ✻ .. 1 x .. ... ... .... ....... ................. ..................... ....... .... ... ... .. B. f 1 n+r ✲ ¨ = r f¨ ur alle r ∈ [0, 1[ (vgl. 15). 3 : Betrachte die Funktion f : R → R, x → 1 x2 +1 . 1. GRENZWERTE VON FUNKTIONEN 35 ✻ ........................................ .......... ............ ........................ . ✲ Wir zeigen, daß limx→1 f (x) = 12 .

An }) := min min({a1 , . . , an−1 }), an und das Maximum einer endlichen Menge {a1 , . . , an } ⊆ Z durch max({a1 , . . , an }) := max max({a1 , . . 5 : Sei (Z, +, ·, P ) ein geordneter K¨ orper. Dann gilt f¨ ur a, b, c, d ∈ Z (i) a < b ⇒ a + c < b + c. (ii) (a < b, c < d) ⇒ a + c < b + d. (iii) (a < b, 0 < c) ⇒ ac < bc. (iv) (a < b, c < d, 0 < b, 0 < c) ⇒ ac < bd. 26 KAPITEL 1. 4. 6 impliziert dies (b + c) − (a + c) = b − a ∈ P also a + c < b + c. (ii) Mit (i) findet man a + c < b + c < b + d, also wegen der Transitivit¨at a + c < b + d.

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