Suppose $X$ and $Y$ are two dependent random variables (e.g. they are the elements of a bivariate normal distribution with $\rho\ne0$). Is it true that there always exists a function $f$ such that $f(X,Y)=0$?
If the statement is not generally true, are there any special cases? Moreover, how the $f$ in question (if any) could be found, given the pdfs of $X$ and $Y$?