In elementary Mathematics, one learns function as a rule for mapping elements $X \to Y$. For example, the rule for a function $f: R \to R$ could be written as:
$$ f(x) = x^2$$
But, as one learns higher mathematics, one learns function just a certain subset of $X \times Y$. Now, if it is just a subset, it may or may not be that the function has a rule to evaluate it element wise. Is there any way to characterize if it does or not? What type of restrictions are needed on the codomain and domain spaces such that all functions between are writable as an evaluation rule?
If $X$ is finite and $Y$ is countable, then all functions $f: X \to Y$ are writable as evaluation rules, since we can express $f$ as a lookup table using finitely many symbols from a finite alphabet.
If $X$ is empty, or $Y$ has one element, then all functions $f$ are the same, so can easily be described.
If $X$ is infinite and $Y$ has at least two elements, or $X$ non-empty and $Y$ is uncountable, then there are uncountably many $f$, and only countably many expressions for $f$, so almost all functions $f$ are inexpressible.
Given a specific $f$, asking whether it can be expressed will depend on your choice of symbol language.