Is there zero factors in analytic function space?

Question

Given real analytic functions $a(x), b(x)$, does $\forall x\in R:a(x)b(x)=0$ implies that either $\forall x\in R: a(x)=0$ or $\forall x\in R:b(x)=0$? Please prove or disprove it.
Well, basically I tried to say that if $a(x)\not= 0$ (zero function), then since $a(x)$ is analytic, its zeros are isolated. Consider its zeros are $U=\{x\in R:a(x)=0\}$, it must be the case that $b(x)=0$ in $U^c=R- U$
and if $U^c$ is (partially) connected, then we will have $\forall x\in R:b(x)=0$ But there're also many cases that $U^c$ is not (partially) connected, for example, if $U=Q$ be the set of rational numbers, then its complement will not be connected.

Answer 1

I think I've known how to solve the problem. Since there's no answer I'll post it right here.
It follows from the fact that being analytic implies continuity.
Assume $a(x)\not=0$(zero function) then there must be some open interval $I$ that $\forall x\in I: a(x)\not=0$. This implies that $\forall x\in I: b(x)=0$, and since $b(x)$ is analytic, $b(x)=0$ is the zero function.