 Differentiability does not imply continuous derivatives

The function $f(x)=x^2\sin (1/x^2)$ for $x\ne 0$, extended by $f(0)=0$, is differentiable everywhere. At zero its derivative is zero by the difference quotient, while away from zero $f^{\prime }(x)=2x\sin (1/x^2)-2x^{-1}\cos (1/x^2)$ is unbounded near zero. Differentiability of a function is therefore weaker than continuity of its derivative.