Graph of a function of multiple variables

Definition

Suppose $f$ is a real-valued function of $n$ variables $x_{1},x_{2},\dots ,x_{n}$. The graph of $f$ is a subset of $\mathbb {R} ^{n+1}$, with coordinate axes $x_{1},x_{2},\dots ,x_{n},x_{n+1}$, given as follows:

$\{(x_{1},x_{2},\dots ,x_{n},f(x_{1},x_{2},\dots ,x_{n}))\mid (x_{1},x_{2},\dots ,x_{n})\in \operatorname {dom} (f)\}$

where $\operatorname {dom} (f)$ denotes the domain of $f$.

Alternatively, it is given by the equation:

$x_{n+1}=f(x_{1},x_{2},\dots ,x_{n})$

For nice enough functions, this graph looks like a hypersurface of codimension one (and dimension $n$) inside $\mathbb {R} ^{n+1}$.