I guess this could be a very elementary question. Anyway I can not find an answer in literature.

Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. Let us write $f(z) = u(x,y)+iv(x,y)$, where $z = x+iy$.

Is there a way of proving that $u,v\in C^{1}(U)$ without using complex integrals, and in particular without using the Cauchy integral formula?

Here holomorphic in $U$ means holomorphic in any $z_0\in U$ that is for any point $z_0\in U$ there exists the limit $$f^{'}(z) = \lim_{z\mapsto z_0}\frac{f(z)-f(z_0)}{z-z_0}.$$