# Increasing function

A real-valued function $f$ defined on a certain set $E$ of real numbers such that the condition

$$x ^ \prime < x ^ {\prime\prime} ,\ \ x ^ \prime , x ^ {\prime\prime} \in E$$

implies

$$f ( x ^ \prime ) < f ( x ^ {\prime\prime} ).$$

Such functions are sometimes called strictly increasing functions, the term "increasing functions" being reserved for functions which, for such given $x ^ \prime$ and $x ^ {\prime\prime}$, merely satisfy the condition

$$f ( x ^ \prime ) \leq f ( x ^ {\prime\prime} )$$

(non-decreasing functions). The inverse function of any strictly increasing function is single-valued and is also strictly increasing. If $x _ {0}$ is a right-sided (or left-sided) limit point of the set $E$( cf. Limit point of a set), if $f$ is a non-decreasing function and if the set $A = \{ {y } : {y = f( x), x > x _ {0} , x \in E } \}$ is bounded from below — or if $\{ {y } : {y = f( x), x < x _ {0} , x \in E } \}$ is bounded from above — then, as $x \rightarrow x _ {0} +$( or, correspondingly, as $x \rightarrow x _ {0} -$), $x \in E$, the values $f( x)$ will have a finite limit; if the set is not bounded from below (or, correspondingly, from above), the values $f ( x)$ have an infinite limit equal to $- \infty$( or, correspondingly, to $+ \infty$).

If $f$ is non-decreasing on $E$ and $x _ {0} \in E$, then the set $A$ referred to above is automatically bounded from below by $f ( x _ {0} )$, unless it is empty. If, in addition, $x _ {0}$ is a limit point of $\{ {x \in E } : {x > x _ {0} } \}$, then the right-hand limit of $f$ at $x _ {0}$ is simply the infimum of $A$:
$$\lim\limits _ {x \downarrow x _ {0} } f ( x) = \inf A .$$