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.