Question

Show that a complex function f(z)=|z|2 is continuous on all complex plan c, but it is only differentiable at the origin.

Answer 1

Answer:

f(z)=x2+y2+i⋅0=u(x,y)+iv(x,y), where u(x,y)=x2+y2 and v≡0. The functions u and v are continuous, so is f. But Cauchy-Riemann only holds at the origin:

ux=vy,uy=−vx⟹2x=0,2y=0⟹z=0,

so since f is continuous, f is differentiable only at the origin, and the derivative is zero.