1. ## Continuity Question

In a few proofs that I have read in Complex Analysis, it uses a property that says:

If a function f is continuous, then it can be express as $f(z)=g(z)+h(z)$

Where $h(z)$ shrinks to zero. Now I'm not sure if the statement is 100% correct, but why is it true?

Thank you.

2. What are the conditions on g(z)? otherwise you could just make $h(z)\equiv 0$ and $g(z)\equiv f(z)$ and the continuity of f(z) is irrelevant.