Aka: Yes, you can still use Cauchy’s theorem if a function f is holomorphic on the interior of a domain and continuous up to the boundary of (at least when is a domain with smooth boundary). Or as Milton pointed out, in the specific question that concerns you, it is easy to see that you can apply Cauchy’s theorem: you can use the uniform continuity of f on the closed unit disk to show that the integral on the unit circle can be approximated by the integral on a slightly smaller circle.