Let $(X, d)$ be a compact metric space and $f:X\to \mathbb{R}$ is continuous on a comeager set $A\subseteq X$. Choose $p, q\in A$ with $f(p)\neq f(q)$.
Is it true that there are open sets $U$ of $p$ and $V$ of $q$ in $X$ such that $f(U)\cap f(V)=\emptyset$?
Note that a set is meager if it is countable union of nowhere dense sets and it is comeager if it is complement of meager set.