Differentiability implies continuous derivative?

Question

We know differentiability implies continuity, and in 2 independent variables cases both partial derivatives fx and fy must be continuous functions in order for the primary function f(x,y) to be defined as differentiable.

However in the case of 1 independent variable, is it possible for a function f(x) to be differentiable throughout an interval R but it's derivative f ' (x) is not continuous?

http://calculus.subwiki.org/wiki/Derivative_of_differentiable_function_need_not_be_continuous — sranthrop, Jan 28 '15 at 13:18

score 0 · Answer 1 · answered Jan 28 '15 at 13:20

0

The derivative can be very poorly behaved, for instance if may even fail to be Riemann integrable. See for instance, the Volterra function.

answered Jan 28 '15 at 13:20

Umberto P.

52,165

Differentiability implies continuous derivative?

1 Answers1