Definition talk:Graph of Real Function

We've got Real Function defined as a mapping whose domain and codomain are $\R$ rather than $\R^m$ and $\R^n$ and so on. This is how it is defined in the sources I have available to me. However, I am wondering more and more how essential it is that the a "Real Function" should in fact be constrained to be one-dimensional.

Obviously in the context of manifolds, where the multidimensionality of the domain and codomain are of great importance, this makes immediate sense, but there's probably a danger of confusing those of a less advanced stage.

My gut feeling is to refer to such multidimensional "real functions" just as mappings like everything else which is not a "real function" or a "real-valued function" or "complex function" etc., which would make it perhaps better to subsume the the page "Graph of Real Function" into "Graph of Mapping", and reserve the name "Graph of Real Function" to the specific case where the codomain is genuinely just $\R$.

It's all the same in the end: it's the subset of the cartesian product which defines the correspondence between the domain and codomain, whatever the latter are. --prime mover (talk) 22:11, 12 April 2021 (UTC)
 * Fine with me. We can have it under graph of mappings. We also have Definition:Real Multivariable Function and Definition:Vector-Valued Function, so maybe we should call it Graph of Real Multivariable Vector-Valued Function? Extra words should signal that this is not just a simple function everybody is used to.--Julius (talk) 06:16, 13 April 2021 (UTC)


 * Perhaps the bigger question is: why bother to have a separate definition at all? It's just the graph of a mapping applied to a specific domain. Perhaps make it a subpage or an example of a graph of a mapping. IMO it's important to stress that all these things are the same sort of thing.


 * One reason for having a separate page for the graph of a real function $f: \R \to \R$ or $f: S \to \R$ where $S \subseteq \R$ is that you can conveniently present a picture of it as the "canonical" graph that is presented to children in grade school -- hence providing the link to a page which is understood at elementary level -- but by the time you get to manifolds, you are probably going to be familiar with general mappings. --prime mover (talk) 06:32, 13 April 2021 (UTC)


 * Ok, we can transclude it as an example. Still, it should be worthwhile providing a separate title just to avoid calling it an "example of graph of mapping". One more reason for the more explicit presentation was the fact the graph of mapping involves a relation which may be a bit too abstract for beginners.--Julius (talk) 08:45, 13 April 2021 (UTC)


 * When you're on the level of manifolds it is unlikely that you're a such a beginner for graph of mapping to be too abstract. Basic elementary real analysis like pre-calculus students get introduced to, yes maybe, but when you're on the level of manifolds you'll already have been through at least a modicum of topology, in which the concept of a general mapping is taken for granted. So I'd revisit the idea of having a separate page for the graph of a mapping from $\R^n$ to $\R^k$.


 * And if "graph of mapping" is too abstract on the grounds of it referring to a general relation, then maybe there is a case for adding a definition which does *not* use a relation. I don't know, myself, I think the concept of the "graph of a mapping" is unwieldy and cumbersome to the point of uselessness -- but it's how some sources develop the subject and so if that's how they roll, that's how they roll. --prime mover (talk) 18:44, 13 April 2021 (UTC)


 * If Lee *does* insist on treating this "graph of function" in this format (in the context of manifolds *everything* is $\R^m$ to $\R^n$ from my understanding) then we could definitely do with a redirect to "/Example", but "Graph of Real Multivariable Vector-Valued Function" is very unwieldy. --prime mover (talk) 19:08, 13 April 2021 (UTC)


 * I added this definition simply because I was trying to write out the definitions one-by-one as I see them. However, now I see that not everything will be that straightforward. In this case, Definition:Graph of Real Function is expendable, i.e. we can simplify it to 1D and hide the rest under /Example/Vector-Valued/Multivariable or any other tag. If I see that further work refers to it, then we can adjust. However, I am planning to spend some time on manifolds, at least the definitions since I need to rehearse all this stuff for my thesis. And as far as I see, multivariable calculus is underdeveloped here. Therefore, there may be more of such multidimensional generalizations skipping 1D case.--Julius (talk) 21:14, 14 April 2021 (UTC)


 * Fair enough. If something looks like it could do with refactoring, slap a template onto it and flag it for future development. The best is the enemy of the good. Just post stuff then, get the material in place, then we can go through it and clean up later. --prime mover (talk) 21:42, 14 April 2021 (UTC)