real analysis - Show $h$ is continuous on $D$

Suppose $f,g:D→\mathbb R$ are both continuous on $D$. Define $h:D→\mathbb R$ by $h(x)=\max[f(x),g(x)]$. Show $h$ is continuous on $D$.

This question is already listed twice in other places and those explanations are from 3 years ago. Not saying math has changed, but I'm still kind of confused after reading them and I know commenting on them won't bring new attention.

I understand that either $f,g$ are equal or one is greater than the other, but I don't know what that does. I imagine that $h$ could be some kind of wild function with points fluctuating and never actually touching or creating a smooth line. But, we are looking for continuity on it's own domain.

Answer

Write
$$\max(f(x),g(x))=\frac{f(x)+g(x)+|f(x)-g(x)|}{2}$$
But if $f$ is continuous at $a\in D$, then also $|f|$ is continuous at $a$, only see
$$\big| |f(x)|-|f(a)| \big| \leq |f(x)-f(a)|$$