I would like to show that $\operatorname{Arg}z$ is not an analytic function by using the Cauchy–Riemann differential equations.
After a quick search I've found this:
From Wikipedia: Alternatively, the principal value can be calculated in a uniform way using the tangent half-angle formula, the function being defined over the complex plane but excluding the origin:
$ \operatorname{Arg}(x + iy) = \begin{cases} 2 \arctan \left( \frac{y}{\sqrt{x^2+y^2}+x} \right) & \qquad x > 0 \text{ or } y \ne 0 \\ \pi & \qquad x < 0 \text{ and } y = 0 \\ \text{undefined} & \qquad x = 0 \text{ and } y = 0 \end{cases} $
Unfortunately, I was not able to make further progress. My question: Is it possible to show that $\operatorname{Arg}z$ is not an analytic function by using the Cauchy–Riemann differential equations?