In what manner are functions sets? From Introduction to Topology, Bert Mendelson, ed. 3, page 15:

A function may be viewed as a special case of what is called a relation.

Yet, a relation is a set

A relation $R$ on a set $E$ is a subset of $E\times E$.

while a function is a correspondence or rule.  Is then a function also a set?

Functions correspond to an abstract rule. Not to something like $f(x)=x+3$. This abstract rule need not be expressible, or even something that you can imagine. Functions, just like any other mathematical object, can be represented as a set. For example, real numbers can be thought of as sets.
Functions are represented as sets of ordered pairs. When we say that $f$ is a function from $X$ into $Y$ then we mean to say that $f$ is a set of ordered pairs $(x,y)$ such that $x\in X$ and $y\in Y$, and the following holds:


*

*For every $x\in X$ there is some $y\in Y$ such that $(x,y)\in f$.

*If $(x,y)\in f$ and $(x,y')\in f$ then $y=y'$.


When the latter occurs we can simply replace the $y$ by $f(x)$.
For example, $\{(0,0),(1,0)\}$ is a function from $\{0,1\}$ into $\{0\}$. 
