Prove that if $f$ is an holomorphic function with certain conditions then $f=0$

Question

Prove that if $f$ is an holomorphic function on $H=\{z\in \mathbb{C} : \, Im(z)>0\}$ such that $ \exists a,b\in \mathbb{R} $ with $a<b$ and $\displaystyle\lim_{z \to x}f(z)=0\quad \forall x\in (a,b)$ then $f(z)=0$, $z \in H$.

Any idea about how to start?