Can Cauchy's integral formula be used on the boundary point?

Question

Can Cauchy's integral formula be used if the point a is on the boundary of the circle ? The theorem is given below.

Answer 1

No. The boundary of the circle is $\gamma$, and if $a\in\gamma$, then the integrand doesn't exist at $a$.

If you need to work with an $a$ on the boundary of $D$, then enlarge $D$ a little. You can do that because $\overline{D}\subset U$.