I'm given the following scenario: Letting $U$ be an open subset of $\mathbb{R}^n$ and $f,g:U\rightarrow \mathbb{R}$, two smooth functions such that $f(\vec x)\lt g(\vec x)$ for all $\vec x\in U$, we'll consider the set $M$ of all pairs $(\vec x, y)$ such that $x\in U$ and $y\in [f(\vec x),g(\vec x)]$. I want to know that $M$ is a manifold with boundary under these conditions, in particular the dimension and what the boundary is. In the event that the definitions of manifold with boundary differ, I'm using the following:
An n-dim manifold with boundary in $\mathbb{R}^N$ is a subset M such that for all $\vec x \in M$ there exists and open subset $V\subseteq \mathbb{R}^N$ containing $\vec x$, an open subset $U\subseteq \mathbb{R}^n$, and an embedding $\psi:U\rightarrow \mathbb{R}^N$ such that $\psi(U\cap H^n)=V\cap M$ (where $H^n$ is the usual upper-half space).
So I'm thinking I can cover the manifold with two embeddings namely $\psi_1(\vec x, z)=(\vec x, f(\vec x)+z)$ and $\psi_2(\vec x, z)=(\vec x, g(\vec x)-z)$. Am I on the right track here? Certainly the graphs of f and g are part of the boundary but is there more? And more generally, what can go wrong in the event the $f\lt g$ condition isn't satisfied? As always, if I've poorly articulated anything, lemme know!