I'm reading Do Carmo's book on curves and surfaces and in chapter 2.3 he claims that $x^{-1}$, where $x$ is a parameterization of a regular surface is differentiable without given a reason.
We know from the theorem about a change in parameterization that for any two parameterizations x and y, $h = x^{-1} \circ y$ is a diffeomorphism, however, we cannot conclude from this that $x^{-1}$ is differentiable since we don't know that y is a diffeomorphism. Can someone tell me why the inverse of x is differentiable?