# Monotone

Monotonicity is a property of functions.

A function $f$ is monotone if and only if when $x \le y \implies f(x) \le f(y)$, for all $x$ and $y$ in the domain of $f$.

A function $f$ is strictly monotone if and only if, for all $x$ and $y$ in the domain of $f$, $x \lt y \implies f(x) \lt f(y)$. All strictly monotone functions are monotone, but not vice versa.

A function $f$ is called a strictly monotone ordinal function if and only if it is strictly monotone, its domain is an ordinal number, and its range is a subset of the ordinals.

## Properties

All strictly monotone functions are injective.

If $f$ is a strictly monotone ordinal function, then $x \le f(x)$ for any $x$ in the domain of $f$.

If $f$ provides an order-isomorphism between an ordinal and a subset of the ordinals, then $f$ is strictly monotone.

## Examples of Monotone functions

The identity function is an example of a monotone function that is not strictly monotone.

The aleph function is a strictly monotone ordinal function.