Can a continuous function have a non-continuous derivative? If $f(x)$ is continuous throughout its domain, can its derivative, g(x) be non-continuous for any point?

If so, what effect does a non-continuous derivative have on the function?
For $x\in [0,1]$, $f(0)=0$ and $f(x)=x^2\sin\frac{1}{x}$ for $x\neq 0$.
Then $f(x)$ is differentiable at any point in $[0,1]$, but $f'$ is not continuous at $x=0$.
Yes, consider the function $$f(x) = \begin{cases} 0 \qquad x\leq0 \\ x \qquad x&gt;0\end{cases}.$$

It's gradient is defined everywhere except $x=0$

$$f^\prime(x) = \begin{cases}0 \qquad x&lt;0\\1\qquad x&gt;0\end{cases}.$$

If you want a function which is continuous but no-where differentiable Brownian Motion/Wiener Process satisfies this.