If $\phi$ is a differentiable bijection from [a,b] to [c,d], is the derivative continuous?

Question

If $\phi$ is a differentiable bijection from [a,b] to [c,d], is the derivative continuous?

I have seen a counterexample in the case where the domain and range are unbounded intervals such as the positive reals. Like here:

Does there exist a bijective differentiable function $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$, whose derivative is not a continuous function?

But I was wondering if the statement is true for differentiable bijections that maps compact set to compact sets (specifically closed intervals). If not, is there a counterexample?