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 $).

Comments

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 $:

$$ \lim\limits _ {x \downarrow x _ {0} } f ( x) = \inf A . $$

Similar remarks hold for left-hand limits.