Question

1) Is the function f(z)={[Rez-lmz]²/|z|² ,z≠0 ;0 ,z=0 Continuous at z=0?

Explain the problem with step by step process?

Answer 1

If we let ∆z go to 0 along the line (∆x, 0), the limit is 1. Along the line (0, ∆y), the limit is 0. Since the answers are different, the limit does not exist and f is not differentiable. (23.1b) Using the Cauchy-Riemann equations, show that f(z) = z − z¯ is not differentiable for any z.