I got this question:

Let $f$ be a continuous function on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, If $f'$ is bounded on $(a,b)$, Must it be the case that $f'$ is continuous on $(a,b)$?


  • $\begingroup$ Try something like $x^2 \sin(1/x)$. $\endgroup$ – user99914 May 23 '14 at 11:58

Consider $$ f(x)= \begin{cases} x^2 \sin \frac{1}{x} &\hbox{for $x \neq 0$}\\ 0 &\hbox{for $x=0$} \end{cases} $$ on the inteval $[-1,1]$.

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  • $\begingroup$ Nice counter example. Thanks. $\endgroup$ – MathNerd May 23 '14 at 12:08

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