I am trying to find where the function $f(x+iy)=|x|+i|y|$ is differentiable and then determine where it is analytic.
I have been using the Cauchy-Riemann equations to show where a function is differentiable. I first tried to determine $$\frac{\partial}{\partial x}\left(|x|\right)$$ via the definition of differentiability. I considered $f(z)=|\operatorname{Re}(z)|$, so \begin{align} \lim_{h\to 0} \frac{f(z+h)-f(z)}{h}&=\lim_{h\to 0} \frac{|\operatorname{Re}(z)+\operatorname{Re}(h)|-|\operatorname{Re}(z)|}{h} \\ &\leq\lim_{h\to 0} \frac{|\operatorname{Re}(h)|}{h} \\ &=\lim_{(u,v)\to (0,0)} \frac{|u|}{u+iv} \end{align} But I don't think the approach I am taking is correct. A hint/solution on how to solve this question would be very helpful.
It's sufficient to consider $x$ and $y$ in four quadrant. In the first and third quadrant you have $xy>0$ and the function will be $f(z)=z$ or $f(z)=-z$ which are analytic. In the second and fourth quadrant you have $xy<0$, so the function will be $f(z)=\bar{z}$ or $f(z)=-\bar{z}$ which aren't analytic. Simply you can consider what happens on axis.