I am stuck on the following problem from a previous Functional Analysis grad course that I'm taking to get a better grip on analysis.
Let $I=[-M,M]$ and $h$ a continuous function supported on $I$. Define: $g(z) = \int_I \frac{1}{t-z}h(t) dt$
With this setup: $a)$ Prove $g$ is analytic on $\mathbb{C}\backslash I$
$b)$ Calculte $\lim_{z \rightarrow \infty} zg(z)$
$c)$ For $\epsilon>0$ and $\theta \in \mathbb{R}$ calculate $\lim_{\epsilon \rightarrow 0}g(\theta + i \epsilon) - g(\theta - i\epsilon)$
I managed to solve $a)$ by applying Morera and Fubini noting that $f(t,z) = \frac{h(t)}{t-z}$ is bounded on any triangle outside $I$.
As for $b)$ I've been thinking for a bit, I noticed that the given function is a convolution so I attempted to show that one of the terms approximates the identity, but I'm having problems rewriting the expression to look like an approximate identity problem.\ I moved on to Cauchy's integral formula since I know it is analytic as $z\rightarrow \infty$ but that also did not bear any fruit.
I have not attempted $c)$ yet.
I appreciate any tips on how to approach a problem like this because I don't have any other ideas.