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?

This is false in general. There are differentiable functions with discontinuous derivative. — Ivo Terek, Mar 10 '20 at 05:45
See this: https://math.stackexchange.com/questions/112067/how-discontinuous-can-a-derivative-be — PinkyWay, Mar 10 '20 at 08:00
Also: https://math.stackexchange.com/questions/292275/discontinuous-derivative — PinkyWay, Mar 10 '20 at 08:01
And, when searching for such a function, whose derivative is discontinuous, you might find it among the first derivatives of functions whose second derivative is discontinuous. — PinkyWay, Mar 10 '20 at 08:05
I think the question is incomplete. Please have a relook at it. — user159888, Mar 10 '20 at 08:29

score 1 · Answer 1 · answered Mar 10 '20 at 08:10

1

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

answered Mar 10 '20 at 08:10

PinkyWay

4,565

Well just choosing any discontinuous function won't work. You have to choose a function with essential discontinuity. Note that the integrated function may be not be differentiable at points where original function is discontinuous. Also see this answer: https://math.stackexchange.com/a/3516923/72031 – Paramanand Singh Mar 10 '20 at 10:25
And this answer https://math.stackexchange.com/a/3518190/72031 – Paramanand Singh Mar 10 '20 at 10:29
@ParamananSingh, thank you for the remark and links! I'll examine them carefully. – PinkyWay Mar 10 '20 at 10:30

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

1 Answers1