I vaguely remember a question going something like
Let $f$ be a function on $[-1,1]$ with $f$ satisfying (something like) $$f(x^2-1)=(2x)f(x).$$ Show that $f$ is identically zero on $[-1,1]$.
Sorry if I can't give much information. The exact statement has been bugging me for sometime now. I'd like to know what the exact statement is.
Edit: Swapped arguments.
Edit: Replaced $$f(x^2-1)=(2x-1)f(x).$$ with $$f(x^2-1)=2xf(x).$$.