# Continuity of injective derivative of a differentiable function

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 $(t,u)$.
Derivatives satisfy the Intermediate Value property by Darboux's Theorem. So, $h'$ is a monotone function that has the Intermediate Value property. So ...