How do you prove that for $|x−x_0|<\delta \Rightarrow |f'(x)−f'(x_0)|<\epsilon$.

Question

How do you prove that for $|x−x_0|<\delta \Rightarrow |f'(x)−f'(x_0)|<\epsilon$, when $g'(x)$ is continuous.

I know that $|x−x_0|<\delta \Rightarrow |f(x)−f(x_0)|<\epsilon$, but I'm seeing proofs that require it works for the derivative of f as well. How does that work?

Answer 1

Hint: Think of a discontinuous function, and integrate it. Then you'll find a counter-example.