Suppose $f$ is continuous on $A = \{z : \text{Im}(z)\geq0\}$ and analytic on $\Omega=\{z : \text{Im}(z) > 0\}$. Let $T$ be a triangle in $A$ with one side on the $x$-axis. Prove that $\int_Tf(z) dz = 0$.
I am totally stuck on how to do this. I know there will be some approximation to $T$ be triangles in the upperhalf plane.