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We know that every equation has a graphical representation by a curve, but does every curve have an equation? If I scribble something crazy on a coordinate plane, do we know if there's an equation that can model my "graph"? Is there any limit to the types of graphs we can create?

I started wondering about this during my experimentation with implicit equations in graphing programs. The more complex an equation is, the wackier its graph can look. You can find various samples of implicit equations here: https://www.desmos.com/calculator/nbbfooa6ei

So if I scribbled something random, the equation would arguably be exorbitantly long, if it existed. Such a drawing would be within a definite set of boundaries for x and y. Most types of equations extend to infinity on the coordinate plane, however equations like x^2+y^2=1 prove that that is not always the case. So can a single equation be conjured for any type of graph? Is there any graph which we can prove has no equation? Piecewise-defined functions don't count here.

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    $\begingroup$ What do you mean by an equation? Under a reasonable definition, the answer is no: there are too many graphs (more than real numbers) but only as many equations as there are reals. But under another definition the answer is yes, trivially, although the "equations" would be infinitely long. $\endgroup$ Aug 24 '16 at 19:35
  • $\begingroup$ I wish you will pardon me for the (proposal of) replacement of "graph" (at least in the title) by "curve". Explanation : there is a quite different concept named also "graph" in mathematics : have a look for example at (en.wikipedia.org/wiki/Graph_theory). And a large majority of people doing mathematics now think, when they see the word "graph", think to this second meaning. $\endgroup$
    – Jean Marie
    Aug 24 '16 at 19:57
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    $\begingroup$ In the present context (that of implicit/parametric curves) there's really no possibility for confusion: Graphs, in the sense of graph theory, are discrete structures whereas smooth curves are certainly continuous. $\endgroup$ Aug 24 '16 at 19:59
  • $\begingroup$ @tphilli Do you only want to approximate the curve, or actually find the equation fitting exactly ? $\endgroup$
    – Peter
    Aug 24 '16 at 20:18

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