Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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This problem introduces a very reduced version of the inverse function theorem.

Suppose \( f : \mathbb{R} \to \mathbb{R} \) is continuously differentiable. Show that if \( f'(x_0) > 0 \) for some \( x_0 \in \mathbb{R} \), then there exists some interval \( I = (x_0 - \delta, x_0 + \delta) \) such that \( f|_I : I \to f(I) \) is bijective.
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Transcribed Image Text:Suppose \( f : \mathbb{R} \to \mathbb{R} \) is continuously differentiable. Show that if \( f'(x_0) > 0 \) for some \( x_0 \in \mathbb{R} \), then there exists some interval \( I = (x_0 - \delta, x_0 + \delta) \) such that \( f|_I : I \to f(I) \) is bijective.
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