NOTICE: Citizendium is still being set up on its newer server, treat as a beta for now; please see here for more.
Citizendium - a community developing a quality comprehensive compendium of knowledge, online and free. Click here to join and contribute—free
CZ thanks our previous donors. Donate here. Treasurer's Financial Report -- Thanks to our content contributors. --

Monotonic function

From Citizendium, the Citizens' Compendium
(Redirected from Monotone sequence)
Jump to: navigation, search
This article is developing and not approved.
Main Article
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

In mathematics, a function (mathematics) is monotonic or monotone increasing if it preserves order: that is, if inputs x and y satisfy then the outputs from f satisfy . A monotonic decreasing function similarly reverses the order. A function is strictly monotonic if inputs x and y satisfying have outputs from f satisfying : that is, it is injective in addition to being montonic.

A differentiable function on the real numbers is monotonic when its derivative is non-zero: this is a consequence of the Mean Value Theorem.

Monotonic sequence

A special case of a monotonic function is a sequence regarded as a function defined on the natural numbers. So a sequence is monotonic increasing if implies . In the case of real sequences, a monotonic sequence converges if it is bounded. Every real sequence has a monotonic subsequence.