Why real valued harmonic functions are holomorphic.

Question

Let $f$ be a real valued harmonic function on $C,$ then Claim $g= \frac {\partial f}{\partial x} - i \frac {\partial f}{\partial y} $ as holomorphic and $h= \frac {\partial f}{\partial x} + i \frac {\partial f}{\partial y} $ as need not be holomorphic.

My Attempt

To prove g as holomorphic, we need to satisfy two conditions

1. C.R Eqns : $u_x = \dfrac {\partial ^2 f} {\partial ^2 x} = v_y = -\dfrac {\partial ^2 f} {\partial ^2 y}$ ($\because$ f is homomorphic )

2. $u_x, u_y, v_x, v_y$ exists and continous.

Here existence is guaranteed BUT HOW TO VERIFY THE CONTINUITY OF $u_x, u_y, v_x, v_y$

Answer 1

Since $f$ is harmonic, $f \in C^2$ by definition, so your function is automatically smooth enough to Cauchy-Riemann's equations (in the "other direction".)