If $f$ is a function defined on $[a,b]$ and is differentiable on $[a,b], f'$(derivatives) is continuous

Question

Proposition A: If $f$ is a function defined on $[a,b]$ and is differentiable on $[a,b], f'$(derivatives) is continuous

Is the above proposition correct or wrong? By theorem 5.12(Rudin, "Principles of Mathematical Analysis"), we know that

If $f'(a) < \sigma < f'(b)$, there is a point $x \in (A,b)$ such that $f'(x) = \sigma $,

but I think this one does not prove that derivatives $f'(x)$ for $x \in [a,b]$ are continuous. Since theorem 5.12 is under the title: The Continuity of Derivatives, I just wrote down proposition A myself and want to know if it is true and why it is if it is true.

If I remember well, after that same theorem, Rudin observes that this fact does not imply continuity. The name is due to the fact that this property is one shared by all continuous functions (and an important one). — pppqqq, Aug 04 '13 at 00:46
The property is called the intermediate value property, and the theorem is Darboux's theorem. — cats, Aug 04 '13 at 00:49
Consider $f(x)=\frac{-x^2}{2}$ for $X\leq 0$; $f(x)=\frac{x^2}{2}$ for $x>0$. Then $f'(x)=|x|$ which is not differentiable at 0 — DBFdalwayse, Aug 04 '13 at 01:08

OR. · Answer 1 · 2013-08-04T01:30:21.570

3

$$f(x)=x^2\sin\left(\frac{1}{x}\right),\text{ for }x\neq0$$ $$f(0)=0$$

edited Aug 04 '13 at 01:30

answered Aug 04 '13 at 00:36

OR.

5,941

If $f$ is a function defined on $[a,b]$ and is differentiable on $[a,b], f'$(derivatives) is continuous

1 Answers1