TITLE: Are two mathematically alike functions equal?
QUESTION [27 upvotes]: Consider the functions $f:\mathbb{R}\rightarrow\mathbb{R}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$ defined by the formulas $f(x)=x^2$ and $g(y)=y^2$ $\forall x,y \epsilon \mathbb{R}$. Is it true that $f=g$ as functions?
My thoughts so far:
Intuitively, yes. Since the two functions are equal at every point where they are defined and are defined on the same points, the are effectively the same function. What concerns me here is the different notation of $x$ and $y$. How does that play into the problem? Are the functions still equivalent?

REPLY [2 votes]: A function is defined by two sets, $E$ and $F$, and a rule that to $x\in E$ make corresponds an unique element $y\in F$. The notation is not relevant so yours $f$ and $g$ are equal.