I am interested in how one could prove that $H(\Omega)\subset \mathscr C^1(\Omega)$ without using the Cauchy integral formula, or any other complex analytic techniques, only real analysis.
Thank you in advance
If $f$ is holomorphic then you can write;
$f(z+h) - f(z) + hf'(z) = h \phi(h)$
where $\phi$ has the property that $\phi(h) \rightarrow 0$ as $h \rightarrow 0$, then; $f(z+h) \rightarrow f(z)$, as $h \rightarrow 0$
Hence $f$ is continuous