Differentiability on the boundary of $[a,b]$

Question

Is it possible to have a function being differentiable on $(a,b)$ and continuous at $a$, but not differentiable at $a$?

Btw, do we actually call a function which is one-side differentiable at the boundary simply differentiable?

Answer 1

Let $f(x)=\sqrt{x}$, $x\in[0,1]$, the one-sided derivative at $x=0$ does not exist.

Answer 2

Yes. If a function $f: \mathbb{R} \rightarrowtail \mathbb{R}$ is differentiable at the point $a \in \text{Int}(\text{dom}(f))$, then it must be continuous at $a$; but its reversal is not true.

Answer 3

An example: $f(x)=|x|$ is differentiable on $(0,b)$ for any positive $b$, but not at $0$.