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I am working on a proof and the hypothesis included the following function:

$h(x)=$ max $\{f(x),g(x)\}$

I do not understand this notation or what this is saying. Could someone give me a formal definition and explain what "max" this function is referring to?

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    Simply the largest of the two (... elements in the set). Sadly, there has been no wide adoption of a dyadic $\max$ operator that would allow one to write $h(x)=f(x)\max g(x)$. – 2017-02-27

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You may as well write $$ h(x)=\begin{cases}f(x)&\text{if }f(x)\ge g(x)\\g(x)&\text{otherwise}\end{cases}$$