Say someone gives you two points on an xy-plane, (0,0) and (2,4). Your job is to find a function that fits this criteria. f(x) could be 2x, x^2, or many other functions. My question is if there are any points, doesn’t have to be just two, that will only have one function that fits its criteria.
Maybe some kind of function that involves irrational numbers…just a thought.
A function can be defined as a set of ordered pairs, no two of which have the same first element.
So if you give me two, or more, points, they already constitute a function by themselves, assuming no two of them have the same first coordinate (but different second coordinates). But any larger set of points that includes all the ones you gave me would also be a function.
That’s if you don’t place any restrictions on what kind of a function it should be—like a continuous function, or an analytic function, or a polynomial, or a function with a particular domain. Did you have any such restrictions in mind?
If the set of points is finite, then there are infinitely many functions that pass through those points.
Now I’m not a mathematician, so I won’t be able to explain the reasoning from that link in layman’s terms. Perhaps someone else who is, can chime in on this later.
It just says that if you have a function fitted at a number of points, you can modify it in any way you like away from those points.
ETA you can easily pass lots of various polynomials, e.g., through your given points; just make them of some high degree (not too low: everybody knows two points determine a line, for instance)
Exactly as Gun says, for a finite number of points, there is an infinite number of functions that pass by those points. So the answer to the OP is nope.
Think of it as a map: you are in one place and want to go to another place. How many paths can you follow? if you don’t put a lot of restrictions (only by roads, can’t pass by people’s windows,…) you have an infinite number of paths.
Not a mathematician by trade, but I’ll take a crack:
If you can pick an arbitrary set of n points and specify a function that includes them, there remain an infinite number of points not included in your function. Pick one. But don’t pick one that has the same x value as any of the original n points, because that would be a dick move. Now with your new point, (x, y) in hand, you can specify a new function that includes the original n points and the new point. If you’re feeling really lazy, you can just define the new function as the old function, except that this new function equals y at x. If you’re feeling more ambitious, you can always create a unique polynomial of degree n or less to include your n+1 points. However you do it, you know it’s different from the original function because it includes the new point and the original one didn’t.
Now you have two different functions that both include your original n points. Since we can always do this, there is never a finite set of points that are included in one and only one function. Moreover, since there are always an infinite number of points we could have picked to define our new function, not only is it not possible for there ever to be only a single function that includes the original n points, there are in fact always an infinite number of such functions. Ok, that last statement is simplifying things overmuch, but it’s truthy.
\blacksquare
@Kyrie_Eleison just described how you can create an infinite number of polynomials, given any finite set of points. And polynomials fit pretty much all of the criteria for “nice” or “well-behaved” functions, so even adding any sort of “well-behaved” restriction, no finite set of points will ever be enough to uniquely specify a function (even a well-behaved function) on the reals.
OK, so what about an infinite set of points? Well, clearly, if we specify a value for every real argument, then our function is unique, because we’ve entirely defined it. We don’t even need to put any constraints at all on our function, if we’re doing that.
What about a lesser (but still infinite) set of points? If we have the constraint that our function is analytic (basically, all of its derivatives exist at all points, even when we look at complex numbers), then it’s not all that hard at all to restrict the function to uniqueness: Specifying its value at every point within a finite interval is enough, as is specifying its value at every rational point.
I’m not sure if specifying a value at every integer point is enough to uniquely define an analytic function. Anyone know for that one?
Certainly not. For example, let f(x) = \sin(\pi x). This function has the value zero whenever x is an integer. The same is true for the function f_2(x) = 0, as well as f_3(x) = e^x\sin(\pi x), just to give a few.
What seems to have been eaten from my earlier post is that there is a way to draw a “natural” (i.e., smooth) curve through a bunch of points, namely, using a draftsman’s spline.
Well one way is to find an analytic function that vanishes at all integers. Then that function as well as any multiple of it can be added to one solution to get infinitely many. Try \sin \pi x. There is also \sin 2\pi x, e^x \sin \pi x. You get the idea.
Ninjaed. I should have read @DPRK before answering.
Yup, I should have seen that counterexample.
What if you include the imaginary plane? I know some seemingly odd things happen with imaginary numbers…
If you include the imaginary plane, then you can consider functions holomorphic on complex domains or the entire complex numbers, which people seem to be doing already. You could regard the complex plane as merely \mathbb{R}^2, but that is not as interesting.
You mean, nice things 
Don’t let Gauss hear you call them imaginary.
Real mathematicians™, unlike their imaginary counterparts, call them complex
More specifically, real mathematicians wouldn’t speak of the “imaginary plane” but of the complex plane, which includes the real axis and the imaginary axis.
An aside: This is why I hate the a, b, c, d, e and ? type of numerical sequence questions as I can take any intger as I want and plot nice polynomial function through points (1,a), (2,b), (3,c), (4,d), (5,e) and (6,?). So my question always is: What is the question?
This is why I don´t like riddles. Any riddle either has an infinite number of possible solutions, or none, but only one is considered “valid”.
This means it’s entirely possible for the riddler to only decide what the correct solution is AFTER the other person gives their answer (the answerer? the riddlee?), thus insuring that the given answer is always considered “wrong”.
I think it should be even possible to mathematically prove this using the ideas discussed above. Consider each clue in the riddle to be a data point, and the riddle solution to be a function that passes through each point. This analogy would have to be made solid, which I suspect could be done, but I don’t know enough math to do so.
You can’t rigorously extend this result to riddles. There are an infinite number of possible functions (in fact, a fairly large infinite number, larger even than the number of real numbers), but only a finite number of possible riddle answers. Since the number of possible riddle answers is finite, it’s always possible to give enough clues to narrow the answer set down to one. A poorly (or cleverly) constructed riddle might not do this, but it’s always possible.
I’m not sure why you say there are finite riddle answers. You can always make the riddle answers longer. (ignoring practical considerations). One way to make it rigorous (perhaps) is to point out that all riddle answers can be written as sentences, and all sentences have a 1-1 mapping with integers. Thus there are as many possible riddle answers as integers (technical term is cardinality).
I think I’d take a different route. I’d argue that you can always form true/false questions about the riddle answer, and these questions have the same cardinality the integers, and can be ordered (by the same logic). You then have a plane with such questions on the X-axis, and true/false on the Y-axis. The riddle clues set certain points, but not all, to be true or false, and the riddle answer has a true/false value for all such questions. I think in this way one could make the comparison rigorous.
Well, given a “riddle”, the correct answer should be defined as one that minimizes some Kolmogorov complexity. If you wanted to work towards a semi-rigorous definition.
Riddle answers are sentences, and there are an infinite number of possible sentences, but that doesn’t mean that there are an infinite number of possible riddle answers. A riddle answer must additionally be some thing or concept that is already known to at least the riddler, and (for a fair one) to the solver, and that is a finite (though very large) set.