# Monotonic function

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In mathematics, a function (mathematics) is monotonic or monotone increasing if it preserves order: that is, if inputs x and y satisfy $x\leq y$ then the outputs from f satisfy $f(x)\leq f(y)$ . A monotonic decreasing function similarly reverses the order. A function is strictly monotonic if inputs x and y satisfying $x have outputs from f satisfying $f(x) : that is, it is injective in addition to being montonic.
A special case of a monotonic function is a sequence regarded as a function defined on the natural numbers. So a sequence $a_{n}$ is monotonic increasing if $m\leq n$ implies $a_{m}\leq a_{n}$ . In the case of real sequences, a monotonic sequence converges if it is bounded. Every real sequence has a monotonic subsequence.