5

I understand mathematically that a function can be differentiable but its derivative might have discontinuities, it's not necessary that the derivative of a differentiable function be continuous.

What I can't wrap my mind around is why? Intuitively, let's say I have a function $f(x)$ which is differentiable everywhere on a closed interval $I$, and its derivative $f'(x)$ has a jump discontinuity somewhere in the interval. Then how would the function $f(x)$ be differentiable at the point where the derivative has a jump discontinuity? Shouldn't the two sides of the limit differ, making it not differentiable everywhere? I feel like if the derivative of a function is continuous except at a certain number of points, then the function itself should be differentiable everywhere except at those points, but this isn't necessarily true, and I would like some sort of intuitive explanation of why that is.

J. W. Tanner
  • 63,683
  • 4
  • 43
  • 88
Riccardo Caiulo
  • 677

1 Answers1

10

If the derivative $f'$ is defined everywhere, then $f'$ cannot have jumps. But $f'$ can have a discontinuity that is not a jump.
You can try to find an example of this of the type: $f(x) = x^a\sin(1/x^b)$ and $f(0)=0$.

See also How to prove that derivatives have the Intermediate Value Property and How discontinuous can a derivative be?

GEdgar
  • 117,296