If a function is continuous and differentiable everywhere is the derivative continuous?

Question

Suppose $f$ is continuous on $[a,b]$ and differentiable on (a,b). Does it follow that $f'$ is continuous on $(a,b)$?

BTW, differentiable implies continuous, so the first condition is not that useful. As to how discontinuous the derivative can be: we have had a pretty comprehensive discussion on this website. — Willie Wong, Dec 11 '13 at 09:45

score 6 · Accepted Answer · answered Dec 11 '13 at 09:46

The function,

$$f(x)=\begin{cases} x^2\sin\frac{1}{x} & \text{ if } x\neq 0 \\ 0 & \text{ if } x= 0 \end{cases}$$

is diffrentiable on $\mathbb{R}$

But,

$$f'(x)=\begin{cases} 2x\sin\frac{1}{x}-\cos\frac{1}{x} & \text{ if } x\neq 0 \\ 0 & \text{ if } x= 0 \end{cases}$$

Is not continuous on $x=0$, since $\lim_{x\to 0}\cos\frac{1}{x}$ is not exist.

score 4 · Answer 2 · answered Dec 11 '13 at 09:39

4

For the function $f(x)=x^2\sin{\frac{1}{x}}$ is continuous at $[-1,1]$ and differentiable at $(-1,1)$ but does not have a continuous derivative.

(The problem is at $x=0$)

answered Dec 11 '13 at 09:39

hrkrshnn

2 Answers2