Increasing function

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

implies

Such functions are sometimes called strictly increasing functions, the term "increasing functions" being reserved for functions which, for such given and , merely satisfy the condition

(non-decreasing functions). The inverse function of any strictly increasing function is single-valued and is also strictly increasing. If is a right-sided (or left-sided) limit point of the set (cf. Limit point of a set), if is a non-decreasing function and if the set is bounded from below — or if is bounded from above — then, as (or, correspondingly, as ), , the values will have a finite limit; if the set is not bounded from below (or, correspondingly, from above), the values have an infinite limit equal to (or, correspondingly, to ).

Comments

If is non-decreasing on and , then the set referred to above is automatically bounded from below by , unless it is empty. If, in addition, is a limit point of , then the right-hand limit of at is simply the infimum of :

Similar remarks hold for left-hand limits.