What are "set-theoretic maps"? Can someone explain to me what is the meaning of “set-theoretic maps”? I've encountered this term in real analysis in $n$ variables.  Specifically, I encountered it in the following statement:

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ function such that $Df(x)$, its derivative matrix, is invertible for every point of domain. Let $U$ be an open subset of the domain s.t. $f$ is one-one. Now, define a "set theoretic function" $g:f(U)\to\mathbb{R}^n$. Then $g$ is a $C^1$ function. 

 A: "Set-theoretic map" just means "function".  The term "set-theoretic" is used to stress that any function between the two sets is allowed, without any further restrictions such as continuity or differentiability.  This usage comes from category theory, where the term "map" can generically refer to morphisms in an arbitrary category, so "set-theoretic map" is referring to a morphism in the category of sets (as opposed to the category of topological spaces, or the category of smooth manifolds, for instance), which is just any old function.
(Other variants on the term "set-theoretic map" used in this way that you might encounter include "map of sets" or "set map".)
A: The word "map" is often used as a synonym for "function". Are you familiar with the concept of a function from one set to another? Here are some examples:
$$f:\mathbb{R}\to\mathbb{R}\;\;\textsf{ where }\;\; f(x)=x^2+1$$
and
$$g:X\to Y\;\;\textsf{ where }\;\; X=\{\star,5\}, \;\;\;Y=\{17,\bullet,\triangle\},\;\;\;\textsf{and}\;\;\;\begin{gather}
g(\star)=\bullet\\
g(5)=\triangle
\end{gather}$$
