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# Monotonic function

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.