A harmonic function \(u\in C^2(\Omega)\) is smooth .
Proof
Let \(\eta_r\) be a smooth bump function with \(\supp \eta_r\subseteq B_r(0)\).
The mean value property implies that \(u(x)=(u\ast\eta_r)(x)\). The smoothness of \(u\) is inherited by the smoothness of \(\eta_r\) since
\[ \partial^\alpha(u\ast\eta_r)=u\ast\partial^\alpha\eta_r. \]