Strictly monotone function

Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function. The function $f$ is strictly increasing if $f(x) < f(y)$ for all $x, y \in D$ with $x < y$. It is strictly decreasing if $f(x) > f(y)$ for all $x, y \in D$ with $x < y$. It is strictly monotone if it is strictly increasing or strictly decreasing.