# 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 ${\displaystyle x\leq y}$ then the outputs from f satisfy ${\displaystyle f(x)\leq f(y)}$. A monotonic decreasing function similarly reverses the order. A function is strictly monotonic if inputs x and y satisfying ${\displaystyle x have outputs from f satisfying ${\displaystyle f(x): 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 ${\displaystyle a_{n}}$ is monotonic increasing if ${\displaystyle m\leq n}$ implies ${\displaystyle a_{m}\leq a_{n}}$. In the case of real sequences, a monotonic sequence converges if it is bounded. Every real sequence has a monotonic subsequence.