Let $f:\Omega\rightarrow \mathbb{C}$ be an holomorphic function i.e. for any $z_0\in \Omega$ there exists the limit: $$f^{'}(z_0) = \lim_{z\mapsto z_0}\frac{f(z)-f(z_0)}{z-z_0}.$$ Let us write $f(z) = u(x,y)+iv(x,y)$. I know how to prove that the partial derivatives exists of $u,v$ exist in any point $(x_0,y_0)\in\Omega$ and that they satisfy the Cauchy-Riemann equations.
How can one conclude that $u,v\in C^1(\Omega)$?
I would like to prove the Cauchy integral theorem using Green theorem. In order to do that I need $u,v\in C^1(\Omega)$.