Here is a question on my complex analysis textbook.
Let $f\in H\big(\{z\in \mathbb{C}:0<argz<\frac{\pi}{4}\}\big)$ and $f\in C^1\big(\overline{\{z\in \mathbb{C}:0<argz<\frac{\pi}{4}\}} \big)$. If $f\equiv 0 $ on $[a,b]\subset \mathbb{R}^{+}$, then $f\equiv 0 $ on $\{z\in \mathbb{C}:0<argz<\frac{\pi}{4}\}$.
I think since the area is special, some integration formula is needed to derive the result. But I don’t know how. Please give me some help and I’d be appreciate it.