Does a differentiable everywhere function have a continuous derivative?
If a function $f$ defined on $[a,b]$ and differentiable everywhere, does it mean that its derivative $f'(x)$ is continuous everywhere on $[a,b]$?
My understanding is 'yes'. Because if there were 'a gap' in $f'(x)$, we could integrate back to $f(x)$ and show that where the gap is, there would be a sharp "angle" where $f(x)$ is not differentiable.
Solution 1:
Not necessarily. The standard example is $$ f(x)=\begin{cases} x^2\sin\dfrac1x & x\ne0,\\0 & x=0. \end{cases} $$ $f'(0)=0$, but $\lim_{x\to0}f'(x)$ does not exist.