First of all I know that this question has been asked already, but I'm looking for a proof simply using the definition of continuity ($\epsilon$, $\delta$)

Suppose $f,g:D \to R$ are both continuous on $D$. Define $h:D \to R$ by $h(x)=$max{$f(x),g(x)$}. Show $h$ is continuous on $D$.

So, there should be two cases. Let $a$ be fixed.

Case 1: $\lvert f(a)-g(a) \rvert >0$

Case 2: $f(a)=g(a)$

I'm not sure what to do from here

I'm having trouble grasping how to carry out proofs regarding continuous functions. If anyone can give me some insight, that'd be much appreciated.