Using the estimation lemma

Question

I have the question:

Prove using the estimation lemma, for a function $f$ which is continuous in some region $D$ that:

$\lim_{r \mapsto 0}\displaystyle\int_{\Sigma}\dfrac{f(z)}{(z-z_0)}\ dz = 2\pi if(z_0) $

Where $\Sigma = z_0 + re^{it},\ \Sigma \subset D$.

My issue is I have no idea how to find an $M$ so that $|\dfrac{f(z)}{(z-z_0)}| < M \ \ \ \forall z \in D$. This seems far too general! Or am I going about this the completely wrong way?

Answer 1

Hint: $$\int_\Sigma \frac{f(z)}{z-z_0}dz-2\pi i f(z_0)=\int_\Sigma \frac{f(z)-f(z_0)}{z-z_0}dz=i\cdot\int_0^{2\pi} (f(z_0+re^{it})-f(z_0))dt.$$