Monotone Real Function

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $E\subseteq\mathbb{R}$ and let $f:E\to\mathbb{R}$.

1. (i) $f$ is nondecreasing on $E$ if $f(x)\le f(y)$ for all $x,y\in E$ with $x\le y$.

2. (ii) $f$ is nonincreasing on $E$ if $f(y)\le f(x)$ for all $x,y\in E$ with $x\le y$.

3. (iii) $f$ is strictly increasing on $E$ if $f(x)<f(y)$ for all $x,y\in E$ with $x<y$.

4. (iv) $f$ is strictly decreasing on $E$ if $f(y)<f(x)$ for all $x,y\in E$ with $x<y$.

5. (v) $f$ is monotone on $E$ if it is nondecreasing on $E$ or nonincreasing on $E$, and strictly monotone on $E$ if it is strictly increasing on $E$ or strictly decreasing on $E$.