Monotonic function

From Citizendium, the Citizens' Compendium

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.

References

• A.G. Howson (1972). A handbook of terms used in algebra and analysis. Cambridge University Press, 115,119. ISBN 0-521-09695-2.