Checking the continuity and differentiability of a function defined on a non-connected set of $\mathbb{R}$

Question

let $f:\mathbb{R}-\{0\} \to \mathbb{R}$ be defined as $f(x)=|x|$.

Is the function continous and differentiable.

I think the function Is continuous because $\lim_{x \to 0^+}|x|=0=\lim_{x \to 0^-}|x|$, we need not worry about $f\left(0\right)$ because it is not defined.

However $\lim_{x \to 0^+} f' \left(x\right) \neq \lim_{x \to 0^-}f' \left(x\right)$,

So it is not differentiable.

Correct me if I am wrong.

Answer 1

You should not worry about $x=0$ because it is not in your domain.

For any other point the function is both continuous and differentiable.