Correlated Implies Non-Zero Conditional Expectation?

Question

Suppose $X$ and $Y$ are correlated, and that they each have mean zero. As such, $E(XY) \neq 0$.

Is it true that $E(X|Y) \neq 0?$

The way I define the conditional expectation $E(X|Y)$ is the function $h(Y)$ such that $$ E\left((X - h(Y)) \cdot g(Y)\right) = 0 \text{ for all functions } g(\cdot). $$

I'm unable to get anywhere with this, even though it seems like a pretty simple question. Can anyone lend a hand?

Answer 1

Hint:

In your definition of $\ E(X|Y)\ $ take $\ g(Y)=Y\ $, giving $\ E\big(h(Y)Y\big)=$$E(XY)\ne0\ $. Is this possible if $\ h(Y)=0\ $?