Give an example of a function \(f:\RR\to \RR\) that is everywhere differentiable but \(f'\) is not continuous at 0.
Solution
Take \[ f(x)= \begin{cases} x^2\sin(1/x),&x\ne0,\\ 0,&x=0. \end{cases} \] At \(0\), \[ f'(0)=\lim_{h\to0}\frac{h^2\sin(1/h)}h =\lim_{h\to0}h\sin(1/h)=0. \] For \(x\ne0\), \[ f'(x)=2x\sin(1/x)-\cos(1/x). \] Thus \(f\) is differentiable everywhere, but \(f'(x)\) has no limit as \(x\to0\) because the term \(\cos(1/x)\) oscillates. Hence \(f'\) is not continuous at \(0\).