Question

Show that f(z) =z+2z bar is not analytic anywhere in the complex plane

Answer 1

Perhaps easiest is to use the Cauchy-Riemann equations. z¯=x−iy, so u=x and v=−y and so ux=1, uy=0, vx=0, vy=−1, so the equations are never satisfied, at any point.

Your technique will also work at any point. Think of the complex derivative in the alternative way: