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](Ordinal.md "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](Order-isomorphism.md "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](Aleph.md "Aleph") function is a strictly monotone ordinal function.