Continuity of derivative

Question

Are all functions differentiable on an interval which don't have continuous derivative, piecewise defined? I mean do all functions differentiable on an interval have continuous derivative if they don't have piecewise definition? I know about $x^2 . sin(1/x)$ but it is defined separately at $0$

Answer 1

Let $f$ be the function defined at $[0,+\infty)\;$ by

$$f\;:x\;\mapsto \lim_{t\to x^+}t^2\sin(\frac{1}{t}).$$

$f$ is not piecewise defined,

$f$ is differentiable at $[0,+\infty)$ but

$f''(0)$ doesn't exist.