Let's consider a relation $R \subset N \times N$, where $N$ is the set of natural numbers.
According to the definition (correct me if I'm wrong) a function is a special case of relation, which satisfies an additional condition that each element of the domain is related to exactly one element of the codomain.
Now, let's assume $R = \{(a,a) | a \in N\}$, so $R$ is basically an equality relation on natural numbers. It's easy to check that it also meets the condition mentioned above. So, in my understanding, it should be a function. Still I was told that it's actually not (and apparently it has something to do with the reference).
So, is it a function, or not? If not, why?
If you have defined equality on a domain and codomain, where the domain is equal to the codomain, then you have a function. It is called an identity function, and is often denoted $I$, $\mathbf I$, or $\operatorname {id}$, or similar.
So:
$$I: X \to X; I(x) = x$$
However, the symbol $=$ is often defined on everything, even things that are "too big" to be sets. So the "relation": $\mathcal{R}(x,y) : x=y$ might work even if the "domain" of $\mathcal R$ is something like "everything", in which case $\mathcal R$ might not be a relation in the typical sense.