In our several variable course we are introduced to something similar to local parametrization of a manifold.The only difference is that the function $\psi:\Omega\to \mathbb R^n$ is defined on a region $\Omega$ rather than on an open set in $\mathbb R^k$.Our definition of region is a bounded set with content of the topological boundary $0$.The definition of parametrized $k$-surface in $\mathbb R^n$ is as follows:
Let $\Omega$ be a region in $\mathbb R^k$ and let $\psi:\Omega\to \mathbb R^n$ be a smooth one-one map such that $\psi^{-1}:\psi(\Omega)\to \Omega$ is continuous and $D\psi(p)$ has rank $k$ for all $p\in \Omega$.Then we say that $\Omega$ is a parametrized $k$ surface in $\mathbb R^n$.For example, $g(r,\theta)=(r\cos\theta,r\sin\theta)$ where $g:\{(r,\theta):0<r<R,0<\theta<2\pi\}\to \mathbb R^2$ then the image of $g$ is a parametrized $k$-surface in $\mathbb R^n$.
So,I want to know what exactly is meant by the derivative at a point on the topological boundary of the region?This is a small point but needs to be clarified.Can someone provide some help?Does the derivative of a smooth map extend continuous to the boundary or are there some exceptions.