Real Analysis – Proof of Continuity for Max Function

Question

Consider the continuous functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$.

Show that $F:\mathbb{R}\rightarrow\mathbb{R}$ with $x\mapsto \max\{f(x),g(x)\}$ is continuous using the $\epsilon – \delta$ definition of continuity.

I know there must be four cases.

If $f(x)\leq g(x)$ and $f(x_0)\leq g(x_0)$ or

if $g(x)\leq f(x)$ and $g(x_0)\leq f(x_0)$ it is easy.

However, assuming $f(x_0)\neq g(x_0)$, what if

$g(x)\leq f(x)$ and $f(x_0)\leq g(x_0)$ or

$f(x)\leq g(x)$ and $g(x_0)\leq f(x_0)$?

For example:

$|f(x)-g(x_0)|$… how do I get from here to $|x-x_0|$?

Answer 1

Hint: The following identity may make the calculation more familiar. $$\max(a,b)=\frac{1}{2}\left(a+b+|a-b|\right).$$