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If $h$ is a real differentiable function defined on $(0,1)$ with a one-to-one derivative, is the derivative continuous on $(0,1)$?

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Yes. First, the derivative is known to satisfy the intermediate value theorem. Therefore it must be monotone. For if, say, $s<t<u$ and $f'(s)>f'(t)<f'(u)$, then $f'$ would take any value between $f'(t)$ and $\min(f'(s),f'(u))$ in each of the two intervals $(s,t)$ and $(t,u)$.

But any monotone function can only have jump discontinuities, and if the derivative has a jump discontinuity at some point, the function is not differentiable there, since its left and right derivatives at that point wil be the two limits (from left and right) of the derivative. This is due to L´Hôpital's rule.

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Derivatives satisfy the Intermediate Value property by Darboux's Theorem. So, $h'$ is a monotone function that has the Intermediate Value property. So ...

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Ah, yes, I keep forgetting the name of this theorem. Thanks for reminding me. It is a curious theorem, so elementary and still relatively unknown. Perhaps because its uses are so limited. –  Harald Hanche-Olsen Dec 14 '12 at 13:39
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