A function f is said to be continuously differentiable if its derivative f 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)={x2sin(1/x) if x00 if x=0

is differentiable at 0, but

f(x)=2xsin(1/x)cos(1/x)

has no limit as x0.