Conditional Expectation of Two Continuous Functions

Question

Given $f(x)$ and $g(x)$ are continuous functions of $x$, does the following equation hold?

$E(f(x)\cdot g(x)|x)=E(f(x)|x)\cdot E(g(x)|x)$.

In particular, if $E(f(x)|x)=0$, can we conclude $E(f(x)\cdot g(x)|x)=0$?

Presumably yes, but I am not sure.

Answer 1

As the comments point out, $E(h(x) \mid x) = h(x)$, therefore, $$E(f(x)\cdot g(x)|x)= f(x) \cdot g(x) = E(f(x)|x)\cdot E(g(x)|x).$$ If you have $f(x) = E(f(x)|x) = 0$, you have $E(f(x)\cdot g(x)|x) = f(x) \cdot g(x) = 0$.