Question on monotonicity and differentiability

Question

Let $f:[0,1]\rightarrow \Re$ be continuous. Assume $f$ is differentiable almost everywhere and $f(0)>f(1)$.

Does this imply that there exists an $x\in(0,1)$ such that $f$ is differentiable at $x$ and $f'(x)<0$?

My gut feeling is yes but I do not see a way to prove it. Any thoughts (proof/counterexample)?

Thanks!

Have you taken inventory of the continuous, a.e. differentiable functions you've encountered...? :) — Andrew D. Hwang, Jul 23 '13 at 11:29

score 1 · Accepted Answer · answered Jul 23 '13 at 11:52

1

The Cantor function satisfies your hypotheses (up to sign) but the derivative vanishes whenever the function is differentiable.

answered Jul 23 '13 at 11:52

Mikhail Katz

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Take the Cantor function and subtract $\frac12 x$ for an example where the derivative is negative and defined almost everywhere, yet $f(1) > f(0)$. – Caleb Stanford Apr 20 '14 at 16:45

score 0 · Answer 2 · answered Jul 23 '13 at 13:10

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As stated in another answer the Cantor function is a counterexample.
You would need to assume differentiability for all $x \in (0,1)$. "Almost everywhere" is not good enough.

answered Jul 23 '13 at 13:10

Uffe J

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Question on monotonicity and differentiability

2 Answers2