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

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

Solution 1:

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.

Solution 2:

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 $x=0$)