Show that max $\{f,g\}$ is continuous Let max $\{f,g\}$ be the function defined by max $\{f,g\}(x)$ = max $\{f(x),g(x)\}$, for $x \in E$ show that max $\{f,g\}$ is continuous.
How do i use the definition of $\epsilon$, $\delta$ to show this provided that $f,g$ are both continuous function.

Hint: For the $\epsilon$-$\delta$ hassle, for all $\epsilon>0$ you get $\delta_f$ and $\delta_g$, and you simply choose $\delta_{\max\{f,g\}}\mathrel{:=}\min \{\delta_f,\delta_g\}$.
Of course, it may be simpler to prove that $x\mapsto |x|$ is continuous and then use the trick by Mario G, but the beauty is in the eye of beholder, right? :-) 
