A real-valued function defined on a certain set of real numbers such that the condition
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 ).
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.
Increasing function. L.D. Kudryavtsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Increasing_function&oldid=11828