Here is another problem from Complex Analysis. I think this is the most common question we ever see in exams.
If $f$ is continuous on a Jordan arc $\gamma$, prove that the function: $F(z)=\int_{\gamma} {f{(\theta})\over\theta-z}d\theta$ is analytic for all $z$ not in $\gamma$.
What I think is I can prove $F(z)$ is differentiable using definition of the derivative at some arbitrary point $z_{0}$, and the continuity of $F(z)$ is trivially hold.
But showing differentiable using definition seems kind of funky to me. I was wondering if anyone have a better way without using the definition of the derivative.