TITLE: Graphs of functions and level sets
QUESTION [5 upvotes]: While going through the first few chapters of my multivariable calculus book, I came across the following:

The graph of a function of two variables is a surface in $\mathbb{R}^3$ and is a level set of a function of three variables. However, not all level sets of functions of three variables are graphs of functions of two variables.

I am finding trouble grasping the notion of this intuitively. Is it actually impossible to find an arbitrary graph that corresponds to a given level set (for the cases given above)?
Could I possibly ask for a concrete example that demonstrates the statement?

REPLY [4 votes]: The statement is a bit vague but  I think this is probably what the author means.
The graph of a function of two variables... is a level set of a function of three variables.  That is, if $f(x,y)$ is a function of two variables, then the graph $z=f(x,y)$ can be written as
$$z-f(x,y)=0\ .$$
If you now define a function of three variables
$$g(x,y,z)=z-f(x,y)\ ,$$
then the above is simply
$$g(x,y,z)=0\ .$$
This is an example of a level set of $g$, that is, it is $g(x,y,z)={}$ constant.
However, not all level sets of functions of three variables are graphs of functions of two variables.  For example, consider
$$g(x,y,z)=x^2+y^2+z^2\ .$$
One of its level sets is $x^2+y^2+z^2=1$, a sphere.  However, this cannot be written in a form where $z$ is a function of $(x,y)$, because many values of $(x,y)$ give two possibilities for $z$.