Here is the problem in question:
When $\Gamma$ is a simple closed contour, the function $G$ defined by $$G(z) := \frac{1}{2\pi i}\int_{\Gamma} \frac{g(\zeta)}{\zeta - z}d\zeta$$ is analytic in the domain enclosed by $\Gamma$ (assuming only that $g$ is continuous on $\Gamma$). Show that the limiting values of $G(z)$ as $z$ approaches $\Gamma$ need not coincide with the values of $g,$ by considering the situation where $\Gamma$ is the positively oriented circle $|z|=1$ and $g(z) = 1/z.$
We have $\frac{1}{\zeta(\zeta-z)} = \frac{1}{z}\left(\frac{1}{\zeta-z} - \frac{1}{\zeta}\right),$ whereupon $G(z) = \frac{1/z}{2\pi i}(I_1 - I_2)$ where $$I_1 = \int_{\Gamma} \frac{1}{\zeta - z}d\zeta, \, \, \, \, \, I_2 = \int_{\Gamma} \frac{1}{\zeta}d\zeta.$$
Clearly, $I_2 = 2\pi i$ irrespective of $z$ while $I_1 = 2\pi i$ due to the enclosed singularity. How are we supposed to obtain meaningful information by taking a limit as $z \to \Gamma$, when $G(z) = \frac{1/z}{2\pi i}(2\pi i - 2\pi i) = 0$ before the limit operator is even applied? Also, why does $G$ just vanish? We should at least get $G(z) = 1/z$ if the counterexample is to be properly criticized, so there seems to be a mistake in my reasoning one way or another.