Is any parametrization of a smooth curve smooth? Can we always find a smooth parametrization of a smooth curve?

Question

I assume that this must be true because the parametrization describes the same object, but I cannot recall a theorem that would state this explicitly.

Answer 1

No. Take, for instance, the curve in $\mathbb R^2$ defined by $\gamma(t)=(t,t)$ ($t\in[-1,1]$). And now take the reparametrization $\eta(t)=\left(\sqrt[3]t,\sqrt[3]t\right)$ (again, with $t\in[-1,1]$).