Let $x \sim N(0, 1)$ be a random variable distributed as standard normal.
I am looking for functions that satisfy $$E[x f(x)] = 0 $$.
In particular, functions whose best approximation is a constant when the underlying distribution is standard normal.
I know that for example any $f(x) = a x^p$ with $p$ an even positive integer and $a$ any real number satisfy the restriction. I think that perhaps any symmetric function around zero satisfies the restriction, but I haven't proved it.
Are there any others? Any hints or related literature on how to find them?
Thanks!