(This is about parametrized curves in $\mathbb{R^n}$.)
In my book, a parameter change is defined as a bijection $\varphi: J \to I$ between intervals of $\mathbb{R}$ s.t. both $\varphi$ and $\varphi^{-1}$ are smooth.
Why do they have to be smooth? I can understand they have to be continuous. What problems would arise if you didn't force them to be smooth (i.e. $\varphi$ is any homeomorphism between $J$ and $I$)?