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Prove the following version of Darboux's Theorem: let f be differentiable in (a, b). Suppose that the two limits f'(a+) = lim f'(x), f'(b–) = lim f'(x) Ia+ both exist and are finite. Show that 1. (Existence of continuous extension) There is a function g(x) E C[a, b] such that g(x)= f(x) for all т€ (а,).

Prove the following version of Darboux's Theorem: let f be differentiable in (a, b). Suppose that the two limits f'(a+) = lim f'(x), f'(b–) = lim f'(x) Ia+ both exist and are finite. Show that 1. (Existence of continuous extension) There is a function g(x) E C[a, b] such that g(x)= f(x) for all т€ (а,).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove the following version of Darboux's Theorem: let f be differentiable in (a, b). Suppose that the two limits f'(a+) = lim f'(x), f'(b-) = lim f'(æ) T>a+ both exist and are finite. Show that 1. (Existence of continuous extension) There is a function g(x) E C[a, b] such that g(x) = f(x) for all x E (a, b). 2. If f'(a+) > m > f'(b–), then there exists c E (a, b) such that f'(c) = m.
Transcribed Image Text:Prove the following version of Darboux's Theorem: let f be differentiable in (a, b). Suppose that the two limits f'(a+) = lim f'(x), f'(b-) = lim f'(æ) T>a+ both exist and are finite. Show that 1. (Existence of continuous extension) There is a function g(x) E C[a, b] such that g(x) = f(x) for all x E (a, b). 2. If f'(a+) > m > f'(b–), then there exists c E (a, b) such that f'(c) = m.
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