Real function/Increasing and decreasing/Definition

Increasing function

Let ${}I\subseteq \mathbb {R}$ denote an interval and let

$f\colon I\longrightarrow \mathbb {R}$

denote a function. Then ${}f$ is called increasing if

$f(x')\geq f(x){\text{ holds for all }}x,x'\in I{\text{ with }}x'\geq x,$

strictly increasing if

$f(x')>f(x){\text{ holds for all }}x,x'\in I{\text{ with }}x'>x,$

decreasing if

$f(x')\leq f(x){\text{ holds for all }}x,x'\in I{\text{ with }}x'\geq x,$

strictly decreasing if

$f(x')<f(x){\text{ holds for all }}x,x'\in I{\text{ with }}x'>x.$