Consider
\[ f(x)=\begin{cases} x^2 \sin(1/x) & \text{if }x\neq 0\\ 0 & \text{if }x= 0.\\ \end{cases} \]This function is everywhere differentiable but the derivative is discontinuous in 0. (See here for a picture).
Consider
\[ f(x)=\begin{cases} x^2 \sin(1/x) & \text{if }x\neq 0\\ 0 & \text{if }x= 0.\\ \end{cases} \]This function is everywhere differentiable but the derivative is discontinuous in 0. (See here for a picture).