I am reading the famous paper of Renyi, entitled "On measures of dependence" (see here1).
He redefined the maximal correlation in a very general form for both discrete and continuous random variables. For any pair of random variables $(X,Y)$ the maximal correlation denoted by $\rho(X;Y)$ is defined as $$\rho(X;Y):=\sup R(f(X), g(Y)),$$ where $R(\cdot, \cdot)$ is the correlation coefficient and the supremum is taken over all non-degenerate measurable functions $f$ and $g$.
On page 445, he showed that when $(X,Y)$ is distributed continuously, there does not always exist functions $f_0$ and $g_0$ which achieve the above supremum. He gave an example which I can not understand.
In his example, he showed that $\rho(X;Y)=1$ by giving a pair of functions $f(X)$ and $g(Y)$ which are equal with probability one. However, he claimed that if there were a pair of functions, say, $(f_0, g_0)$ such that $R(f_0(X), g_0(Y))=1$ this would yield a contradiction. I dont understand how this comes to the contradiction he claimed.
Can any body help?