Strictly monotonic one to one and continuous function

Question

A function on $[a,b]\rightarrow [c,d]$ is continuous, one to one, and strictly monotonic.(which would imply inverse continuous)

Would ir necessary be dufferentiable?

Answer 1

$$f(x)=\begin{cases}2x & x\geq 0 \\ x & \text{otherwise}\end{cases}$$ which is not differentiable in $0$.

Answer 2

$f(x) = \sqrt[3]{x}$ on the interval $[-1,1]$ is a simple counterexample, since it's not differentiable at $x=0$.

But it can be much worse; take for example $f(x) = x + c(x)$ where $c(x)$ is the Devil's Staircase.