Monotone function

A monotone function on an interval $I\subseteq\mathbb{R}$ is a function $f:I\to\mathbb{R}$ that is either nondecreasing (if $x\le y$ implies $f(x)\le f(y)$) or nonincreasing (if $x\le y$ implies $f(x)\ge f(y)$); it is strictly increasing or strictly decreasing if the inequalities are strict whenever $x<y$.

Monotonicity rests on the order axioms for $\mathbb{R}$ and is often proved using derivative sign implies monotonicity . Monotone functions on compact intervals have good integration behavior, for example monotone implies Riemann integrable .

Examples:

• The function $f(x)=x^3$ is strictly increasing on $\mathbb{R}$.
• The function $f(x)=\frac{1}{x}$ is strictly decreasing on $(0,\infty)$.