Consider two real-valued functions $f$ and $g$ and define $h(x)=Max\{f(x),g(x)\}$, the maximum of the values of the two functions $f$ and $g$ at $x$.
What is the usual convention for the domain of $h$?
I can see two possibilities:
- $h$ is defined if and only if $f$ and $g$ are defined, so $\textrm{dom}(h)=\textrm{dom}(f)\cap \textrm{dom}(g)$. The function $x \mapsto (f(x),g(x))$ is defined on $\textrm{dom}(h)=\textrm{dom}(f)\cap \textrm{dom}(g)$, and $(a,b)\mapsto Max\{a,b\}$ is defined on $\mathbb R^2$, so it makes sense with the usual rule for the domain of a composition.
- If at least one function is defined, we ignore the undefined functions. For example, if $h(x)=Max\{x,\frac 1 x\}$, then $h(0)=0$. In the case all functions are undefined and we have a convention for $Max~ \emptyset$, we can use that convention.
What are the advantages and drawbacks of these definitions?
Note: I am asking out of curiosity. I used the maximum of two functions for simplicity, but it could be extended to the maximum of a family of functions. If it is useful for an answer, please feel free to consider arbitrary families of functions.
By definition, $\max(a,b): \mathbb{R}^2 \to \mathbb{R}$ is defined by $$(a,b) \to \begin{cases} a, &\text{if $a\ge b$}\\ b, &\text{if $a< b$}\\ \end{cases} $$ Since the domain of $\max(a,b)$ is $\mathbb{R}^2$, it follows that $$\max(f(x),g(x))$$ is defined if and only if $f(x),g(x) \in \mathbb{R}$.
Therefore the domain of $\max(f(x),g(x))$ is the intersection of the domains of $f$ and $g$.
Thus, of your two candidate definitions, the first one is standard.
If you have an application that requires the second one, define it and give it a name, but don't call it $\max$.