Increasing function
A real-valued function defined on a certain set
of real numbers such that the condition
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implies
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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
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(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
:
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Similar remarks hold for left-hand limits.
Increasing function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Increasing_function&oldid=11828