[j2kun]

> Not even most of the people who studied Computer Science know the difference between a function and the graph of a function.

I am surprised by how often I hear this example. Not because people can't define the graph of a function, but because in usual set theoretic terms[1] a function and its graph are the same thing. They are both the set of points (x, f(x)) where x ranges over the domain of f. This is an equivalence that isn't even made by most mathematicians, as they usually denote the graph a function as Γ(f), implying it is different from f itself as a function. Of course they know the equivalence, but prefer to separate the objects notationally for clarity's sake.

[1]: I'm ignoring the notion of a function in category theory, since CS students who are taught math are primarily taught naive set theory, not category theory. Of course, when you aren't working in a category of sets then the graph of a function doesn't make sense, I think.

[QuantumRoar]

> I am surprised by how often I hear this example.

Huh? That surprises me, too. I didn't know it was so common.

Anyway, I agree that there is an equivalence between a function and its graph. Knowing one implies you know the other. But just because it's equivalent, doesn't mean it's the same. A two-dimensional graph is, after all, a set of tupels in R^2. A function, however, is a mapping from one element in one set to another element in another set.

Mathematically they're very different. I can't draw functions, I can only draw sets. I can't apply sets, I can only apply functions. It makes sense to treat them as different things.

Actually, vector graphics are mathematically specified by graphs of functions, not pixel sets. This makes it possible to zoom in without pixelation, since the function in the graph can be defined analytically (like Bezier curves). If you started out with sets instead of functions, you'd get pixel graphics, since it's not possible to save an infinite set. So the equivalence of a function and its graph holds until you try to implement something.

[j2kun]

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.