I am looking at the following problem from Marsden and Hoffman:
Let $ f ( z ) $ be analytic inside and on a simple closed contour $ \gamma $. For $ z_0 $ on $ \gamma $, and $ \gamma $ differentiable near $ z_0 $, show that $$ f(z_0) = \frac{1}{ \pi i } \text{ P.V. } \int_{ \gamma } \frac{ f ( \zeta ) }{ \zeta - z_0 } \tag{1}$$
The best idea I have now is to think about $ \gamma_{ \varepsilon } $ defined to be the same contour except with the portion inside the ball of radius $\varepsilon $ centered at the singularity $( z_0 ) $ has been removed. This page then seems to prove a similar result by relating this $\text{ P.V. }$ to the mean value of the integrals displaced slightly above and below so that the residue theorem can be applied. Unfortunately, I do not really know how to start solving the problem this way, or if I am even on the right track.
I would be very appreciative for any help at all, especially if you can explain why $ \gamma $ must be differentiable near $ z_0 $.
P.S. Sorry if this is a duplicate. I made sure to look before posting.