Definition 1

A mapping of a set $X$ into the real line is called a real-valued function (or real function) defined on $X$.

By an abuse of language analogous to that mentioned in § 4, no. 2, mappings of $X$ into $\overline{\mathbf{R}}$ will also be called real-valued functions defined on $X$ in this and the following section. Mappings of $X$ into $\mathbf{R}$ will be called finite real-valued functions.

If $f$ and $g$ are two real-valued functions defined on $X$, the relation $f \leq g$ is by definition equivalent to "$f(x) \leq g(x)$ for all $x \in X$;” this relation is an ordering on the set $\overline{\mathbf{R}}^X$ of all real-valued functions on $X$. The set $\overline{\mathbf{R}},^X$ ordered by this relation, is a lattice; for if $f$ and $g$ are any two real-valued functions, the function $h$ defined by $h(x) = \sup (f(x), g(x))$ for all $x \in X$ is the smallest of the real-valued functions on $X$ which are both $\geq f$ and $\geq g$; in accordance with general notation, we denote this function (which is the least upper bound of $f$ and $g$ in $\overline{\mathbf{R}}^X$) by $\sup (f, g)$. Similarly, the function whose value at each $x \in X$ is $\inf (f(x), g(x))$ is denoted by $\inf (f, g)$.

Note that $\sup (f, g)$ is the composition of the mapping

of $\overline{\mathbf{R}} \times \overline{\mathbf{R}}$ into $\overline{\mathbf{R}}$ and the mapping $x \to (f(x), g(x))$ of $X$ into $\overline{\mathbf{R}} \times \overline{\mathbf{R}}$. Similarly for $\inf (f, g)$.

A real-valued function $f$ defined on a set $X$ is said to be bounded above (resp. bounded below) in $X$, if $f(X)$ is a subset of $A'' = [-\infty, +\infty]$ and is bounded above (resp. if $f(X)$ is a subset of $A' = ]-\infty, +\infty]$ and is bounded below). $f$ is said to be bounded in $X$ if it is bounded both above and below, that is if $f(X)$ is a bounded subset of $\mathbf{R}$.

Every bounded function is therefore finite. The converse is false, as is shown by the function $1/x$ on $\mathbf{R}_+^* = ]0, +\infty[$.