Are intervally constant analytic functions always constant?

Question

Does a real analytic function $f:R→R$ being constant in some open interval implies that it is constant in the whole domain?
If so, how to prove it? Otherwise, is there any counter-example?

Answer 1

That $f$ is constant on an open-bounded interval $I$ implies its derivative $f'$, which is also an analytic function, is zero on $I$, an open domain. By the principle of isolated zeros, $f'$ is identically zero on $\mathbb R$, and $f$ is constant.