A First Course in Complex Analysis · Problem 2.5.7

Differentiation — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · limits-complex-functions

Problem

Prove that

\lim_{z \to z_0} f(z) = w_0

if and only if

\lim_{z \to z_0} \operatorname{Re}(f(z)) = \operatorname{Re}(w_0)

and

\lim_{z \to z_0} \operatorname{Im}(f(z)) = \operatorname{Im}(w_0)