Formal definition of function

Question

In set theory, functions are usually defined as subsets of a Cartesian product. However, this way, the functions $f\colon\Bbb R\to\Bbb R$ and $g\colon\Bbb R\to[-1,1]$, defined as $f(x)=g(x)=\sin x$ for every $x\in\Bbb R$, should be equal. However, I think $f$ and $g$ are still distinct objects, even though the values are the same. Am I wrong? Is there some distinction to be made, or $f=g$? If it's the case that the latter is true, then there is no way of assigning a codomain to a function in a unique way, thus the notation $\operatorname{Cod}(f)$ would be pretty much an informal one.

Answer 1

The most formal definition of function which I have encountered so far is given in the Italian book "Analisi Uno" by Giuseppe De Marco: it fully addresses your issue. He writes:

Let $X, Y$ be sets. A function (or application, or map) from $X$ to $Y$ is an ordered triple $(X, Y, G)$, where $G \subseteq X\times Y$, such that: $$\forall x, x \in X \implies \exists! y \in Y: (x, y) \in G. $$ $G$ is called the $graph$ of the function.