I recently posed the following question in a comment section:
If I draw a curve on a piece of paper that passes the vertical line test and exhibits some particular property (e.g. 'Always greater than $0$'), can I assume that there is some corresponding/matching function whose assignment rules can be explicitly written out?
A reputable user responded with:
The problem is with "assignment rules can be explicitly written out" — what are allowable explicit operations/formulas? (polynomials, step functions, trig, exponential, inverses, expressible as a convergent infinite series of such functions, elliptic functions, hypergeometric functions, etc.) But if we're not asking whether the function can be expressed in a certain way, but simply whether it exists, then any set of points (regardless of how weird) in the plane satisfying the vertical line test is the graph of a function $(*)$
Assuming that I interpreted $(*)$ correctly (and accepting that, perhaps, the response was meant to be informal), it seems to me that something like the following claim is being made:
If on a piece of paper you can draw a collection of points that satisfies the vertical line test, then there is a function that contains these points$(\dagger)$
From searching around the forum, I see that there are a lot of technical nuances regarding "definability" (e.g. https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb/44129#44129/) that go way beyond my skillset. Further, the act of "drawing on a piece of paper" is clearly some meta-concept.
Nonetheless, I would like to see if someone could help me understand why $(\dagger)$ is a reasonable claim. Although the idea of "drawing" doesn't seem to be directly reconcilable within the language of set theory, I know that the notion of a graph is (Difference between a function and a graph of a function?).
Is it safe to say that, by convention:
drawing on a piece of paper a collection of points that satisfies the vertical line test $\iff$ graph $G_f$ of some function $f$
?
In which case, $(\dagger)$ is more precisely stated as:
If $G_g$ is the graph of $g$, then there exists a function $f$ such that $f$'s graph $G_f$ is equal to $G_g$?
The proof of which is trivial.
I pose this question because, if my goal is to prove that there exists a function satisfying some property…and this property happens to be something that can be visually identified through drawing (e.g. "$f$ is always greater than $0$), then is a drawn collection of points (satisfying the desired properties) a satisfactory means of proof?
Best Answer
Idealized correspondence:
First, let's get a concrete question out of the way: is there a convention saying that drawing on a piece of paper a collection of points that satisfies the vertical line test corresponds to giving the graph of some function f?
There is relevant convention, often called the Cantor-Dedekind axiom. It is not a formal axiom in the mathematical sense, but rather a thesis, similar to e.g. the Church-Turing thesis on the nature of computation.
Under this convention, a perfectly precise, idealized drawing of a curve that satisfies the vertical line test, performed on the (again, idealized) geometric plane, would indeed correspond to the graph of a function in the set-theoretic sense, i.e. as a subset $S$ of $\mathbb{R}^2$ which intersects every vertical line $\{(x,y) \in \mathbb{R}^2 \:|\: x=c\}$ in exactly one point. From there on, it's easy to realize $S$ as the graph of a function $f$, simply by defining $f(c)$ as the $x$ coordinate of the unique point $(x,y)$ in $S \cap \{(x,y) \in \mathbb{R}^2 \:|\: x=c\}$.
Drawings in proofs:
Unfortunately, the convention discussed above is not particularly relevant: your question concerns collections of points physically drawn using pencil on actual pieces of paper (or chalk on board etc.), not infinitely detailed idealized drawings on ideal geometric planes!
Specifically, you're interested in drawings that occur in proofs that establish the existence of functions exhibiting some particular properties. So to answer your question, we have to grapple with the not entirely straightforward concept of what it means to prove a mathematical statement.
You may know that formal proof constitutes the gold standard of mathematical argumentation: a derivation in which every step has been justified by the rules of inference of formal logic and by the foundational axioms of mathematics. Formal proofs are written in a precisely defined formal language (precise enough that they can be taught even to computers) and do not contain any graphical illustrations.
Formal proofs are cumbersome, and not really necessary: many mathematicians do not encounter them at all, except perhaps in a university course dedicated to Proof Theory. Instead, most proofs in academic mathematics (that appear in journals, that you study in textbooks, etc.) are rigorous informal proofs.
Unlike formal proofs, informal proofs are communicated in free-form natural languages such as English, may freely skip over tedious, repetitive or routine steps, and often employ accompanying illustrations and diagrams. A rigorous informal proof succeeds if it equips trained (and possibly skeptical) members of the intended mathematical audience with enough information that (in principle and given sufficient time) they themselves could construct a formal proof of the required conclusion.
Informal proofs are literary works that appear on pieces of paper (computer screens, chalkboards etc.), and this is the context where you might encounter a "curve on a piece of paper that passes the vertical line test and exhibits some particular property". So, does such a curve constitute a (rigorous informal) proof? The answer is: it depends. I've prepared two case studies to illustrate what I mean.
Case study 1: Is there a continuous function $f: [0,1] \rightarrow [0,1]$ that has countably infinitely many fixed points?
When asked this question, in lieu of a formal proof, I might draw the following curve (please disregard the scale, I took this from my answer to a different Math.SE question).
Based on this drawing, experienced mathematicians will immediately know how to construct some continuous function $f: [0,1] \rightarrow [0,1]$ with the desired property that $f$ has countably many fixed points. If an experienced mathematician asked me the question above in a seminar, I would just draw the curve, they'd mutter something like "oh, of course", and that would be the end of our discussion.
But what if a student asked me the same question in my real analysis class? A student, who may not yet know all the usual techniques that mathematicians commonly use to construct real functions, might not see how to go from my curve to a formal proof answering the question. But the student can ask for further clarification, and using the curve as a guide, I will be readily able to supply additional details, until either
If the dialogue continues for long enough that it results in a formal proof of the existence of a function $g$ exhibiting the required property, the plot$\!^1$ of the resulting $g$ may not even resemble my initial drawing particularly closely. I would normally construct a curve resembling the shaded area using the function $x \mapsto x \sin(x^{-1})$, but if we haven't covered transcendental functions yet, I may opt to use e.g. piecewise defined parabolas instead, or might even replace the parabolas with linear segments for simplicity.
The drawing is an informal proof, and not (necessarily) a definition of a unique function: it is a mnemonic aid that provides enough detail that a competent mathematician could, if necessary, construct a formal argument with the same conclusion. The curve is sufficient as a rigorous informal proof.
Case study 2: Is there a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ that is not differentiable at any point?
This question was settled by K. Weierstrass in a landmark paper in 1872. Weierstrass constructed a function with the required property using fairly sophisticated Fourier-analytic methods. Had Weierstrass merely presented a drawing with the required properties, like the following,
this would not have constituted a rigorous informal proof of the existence of the required function. It's not particularly clear that there is any real function $f$ whose plot looks anything like this, and filling in sufficiently many details is probably not even possible based purely on the drawing.
So, while the drawing above is a helpful illustration, it, by itself, is not sufficient as rigorous informal proof for the existence of a function exhibiting the required property.
In summary:
There is a convention saying that real numbers correspond to points on an ideal geometric line, but it has no direct implications regarding actual, non-ideal curves drawn on paper.
You may encounter such non-ideal drawings as parts of rigorous mathematical proofs asserting the existence of real functions with certain properties. If the curve provides enough detail to convince a competent mathematician that a function with the required properties exists, then it achieves its goal. This is why drawings are acceptable in mathematical proofs.
However, this is not the same as your statement $\dagger$, which asserts that "if on a piece of paper you can draw a collection of points that satisfies the vertical line test, then there is a function that contains these points". Indeed, there is no sense in which mathematical functions can contain points drawn on actual, physical pieces of paper: at best, one can say that they contain the idealized mathematical points of the Cartesian plane. So as stated, $\dagger$ is neither technically precise, nor true.
If you'd like to read more, I discuss the meaning of geometric proofs in an analytic context further in my answer to the following question: Are geometric arguments using infinitesimals valid?
$\!^1$ Here we distinguish between graph and plot: graph always means the ideal, set-theoretic graph, while plot always means the physical artifact produced when you plot a function.