I am new to complex analysis and got completely confused about complex and real differentiability. I know that the complex absolute value function is nowhere complex-differentiable except at zero. But is the complex absolute value function real differentiable? If so, how could I prove this?
This just came up as I am supposed to show that the Wirtinger derivative of $ln |f(z)|$ is itself holomorphic, where f is a nowhere zero, holomorphic function on the open unit disk. I then asked myself why the Wirtinger derivative should even exist, i.e. why this composite is partially differentiable. I thought it could be easy to prove this saying that it is a composition of real-differentiable functions, but then exactly my problem occurred, as I do not know if real-differentiability also holds for $|.|$.