# Derivative of a bounded function

I was wondering for the bounded function $b(t)$ what statements can be made about the derivative of

$f(t)=exp(b(t))$

specifically it would be nice if the derivative $f'$ were bounded.

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Are you assuming $b$ is also differentiable? –  cardinal Jan 25 '12 at 3:11
no, just bounded unfortunately. Is there anything I can assume since b(t) is bounded => f(t) is bounded => f' can only be unbounded for limited periods –  user23607 Jan 25 '12 at 3:48
See the example I gave in a comment to marty's answer. Nothing about boundedness implies differentiability. Therein lies your problem. –  cardinal Jan 25 '12 at 3:54
thanks everyone email from the prof says i can chose b, so this is trivial –  user23607 Jan 25 '12 at 4:34

Only if $b'(t)$ is bounded, since $f'(t) = b'(t) exp(b(t))$.
Try $b(t) = sin(t^2)$.
I was thinking: Try $b(t) = \log(1 + \chi_{\mathbb{Q}}(t))$. –  cardinal Jan 25 '12 at 3:15