Let $$G = \{ z\in \mathbb{C}: 0<\arg z<\frac{\pi}{4}\},\hspace{0.2cm}f \in H(G)\cap C(\bar{G})$$  

If $f(x) = 0$ for all real numbers $x \in [a,b]$ where $[a,b]$ is a interval in $\mathbb{R}$. Prove that $f(z)$ is identically zero in $G$. 

I'm supposed to use Cauchy integral formula but don't know where to start.