An Elementary Question About Differentiability

Question

If $f(x)$ is differentiable at $x = a$, then $f(x)$ must be defined in an open interval including $x = a$.

If $f'(x)$ is differentiable at $x = a$, then $f'(x)$ must be defined in an open interval including $x = a$.

So, $f(x)$ must be differentiable in an open interval including $x = a$.

Is $f(x)$ differentiable in an open interval including $x = a$ if $f(x)$ is differentiable at $x = a$?

Answer 1

No, consider $f : \Bbb R \to \Bbb R : x \mapsto \begin{cases} x^2 & x \in \Bbb Q \\ -x^2 & x \notin \Bbb Q \end{cases}$.