The graph of $f(x) = x$ has one such curve, the graph of $f(x) = 1/x$ has two such curves, the graph of $x^{2}y^{2}-x^{2}-y^{2}=0$ has four such curves, and the graph of $f(x) = \tan x$ has infinite such curves.
Is there a definition of such a number, given an arbitrary curve? I'd like to find this definition to see if there is a way to compute this number given an arbitrary equation.