Monotonic function: Difference between revisions
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In [[mathematics]], a [[function (mathematics)]] is '''monotonic''' or '''monotone increasing''' if it preserves [[order (relation)|order]]: that is, if inputs ''x'' and ''y'' satisfy <math>x \le y</math> then the outputs from ''f'' satisfy <math>f(x) \le f(y)</math>. A '''monotonic decreasing''' function similarly reverses the order. A function is '''strictly monotonic''' if inputs ''x'' and ''y'' satisfying <math>x < y</math> have outputs from ''f'' satisfying <math>f(x) < f(y)</math>: that is, it is [[injective function|injective]] in addition to being montonic. | In [[mathematics]], a [[function (mathematics)]] is '''monotonic''' or '''monotone increasing''' if it preserves [[order (relation)|order]]: that is, if inputs ''x'' and ''y'' satisfy <math>x \le y</math> then the outputs from ''f'' satisfy <math>f(x) \le f(y)</math>. A '''monotonic decreasing''' function similarly reverses the order. A function is '''strictly monotonic''' if inputs ''x'' and ''y'' satisfying <math>x < y</math> have outputs from ''f'' satisfying <math>f(x) < f(y)</math>: that is, it is [[injective function|injective]] in addition to being montonic. | ||
A [[differentiable function]] on the [[real number]]s 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 number]]s. So a sequence <math>a_n</math> is monotonic increasing if <math>m \le n</math> implies <math>a_m \le a_n</math>. | A special case of a monotonic function is a [[sequence]] regarded as a function defined on the [[natural number]]s. So a sequence <math>a_n</math> is monotonic increasing if <math>m \le n</math> implies <math>a_m \le a_n</math>. | ||
In the case of [[real number|real]] sequences, a monotonic sequence converges if it is [[bounded set|bounded]]. Every real sequence has a monotonic subsequence. | In the case of [[real number|real]] sequences, a monotonic sequence converges if it is [[bounded set|bounded]]. Every real sequence has a monotonic subsequence. | ||
A [[ | ==References== | ||
* {{cite book | author=A.G. Howison | title=A handbook of terms used in algebra and analysis | publisher=[[Cambridge University Press]] | year=1972 | isbn=0-521-09695-2 | pages=115,119 }} | |||
Revision as of 02:59, 19 December 2008
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. Howison (1972). A handbook of terms used in algebra and analysis. Cambridge University Press, 115,119. ISBN 0-521-09695-2.