A function $f$ is said to be continuously differentiable if its derivative $f\prime$ exists and is itself a continuous function.
Although the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity. For example, the function
\[f(x) = \begin{cases} x^2 \sin(1/x) & \text{ if } x \ne 0 \\ 0 & \text{ if } x=0 \end{cases}\]is differentiable at 0, but
\[f\prime(x) = 2x \sin(1/x) - \cos(1/x)\]has no limit as $x \to 0$.