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.

Remarks

With $\le$ and $\ge$ in place of $<$ and $>$, we get increasing, decreasing, and monotone functions. A strictly monotone function is injective. For example, $x^3$ is strictly increasing on $\mathbb{R}$, and $x^2$ is strictly increasing on $[0, \infty)$. Mathlib defines strictly increasing maps as StrictMono.

Explanations

L1 human

A strictly increasing function always goes up: a larger input gives a larger output. Its graph rises from left to right, with no flat parts.