For whatever reason, I hear it as a "challenge" trying to show someone they don't understand math.

But again, I would argue that you're ignoring the difference between an object and its representation for the purpose of communication. A function is defined to be a set of tuples (a relation) with some extra properties. These tuples are the input-output pairs of the function, which is exactly how you define the graph of a function. They are literally the same object, regardless of whether you choose to represent them as a picture or a mapping. The choice of language suggests they are different things because you want to treat them differently (and this is a good thing), but when you get down to the definitions they're the same.

Your example of vector graphics is not a counterexample to this because the sets that define those functions are uncountably infinite. When you render them on the screen to have to approximate them by a finite pixel set, but that is unrelated to the underlying function's mathematical nature.

These set-theoretic definitions of functions and relations are standard parts of an introductory course on set theory.