Let $f:\Bbb R \to \Bbb R$ be continuous.

Does $f \geq 0 \forall x$ and $f>0$ for a countable set of points $A$ imply $\int\_a^b f(x) dx >0$ if $A\subseteq (a,b)$?

I've had this doubt. I'm fairly sure this is not true, but I think it would take a pretty weird function to disprove it.

Does anyone know a simple proof/counterexample?