Prove that if $\lim_{z \to z_0}f(z)=w$

Question

Why we need to show is that the functions are all continuous? Can anyone explain this? I would appreciate it if you help.

Answer 1

You want to interchange a function and a limit. Therefore you need continuity. For example for complex conjugation we can show $\lim\limits_{z\to z_0} \overline{f(z)} = \overline{w}$ by the simple line$$\lim_{z\to z_0} \overline{f(z)} \overset{(*)}{=} \overline{\lim_{z\to z_0} f(z)} = \overline{w}.$$ The eqality $\overset{(*)}{=}$ holds if and only if the function $z \mapsto \overline z$, i.e. complex conjugation, is continuous in $z_0$.

Answer 2

Continuity allows you to say $\lim_{z \to z_0}g(f(z))=g(\lim_{z\to z_0}f(z))=g(w).$