I am trying to prove the Cauchy Integral formula for star shaped regions which will use one of the ideas in the notes posted.
On page 2, the last paragraph says that
If, for each fixed $ \epsilon > 0$, we let the width $\delta$ of the corridor go to zero, the contributions to the integral of the two sides of the corridor cancel due to the continuity of $g$.
This seems conceptually right, but how to write it down rigorously? Where does the continuity of $g$ come in?
Can anyone give me any idea on how to write down the explicit formulas of the integral so that I can actually see how to use the continuity of $g$?