Monotonic Functions

One special type of real-valued functions that are of interested to study are known as increasing and decreasing (collectively, monotonic) functions which we define below.

Definition: A function $f$ is said to be an Increasing Function on the interval $[a, b]$ if for all $x, y \in [a, b]$ where $x < y$ we have that $f(x) \leq f(y)$. $f$ is said to be a Decreasing Function on the interval $[a, b]$ if for all $x, y \in [a, b]$ where $x < y$ we have that $f(x) \geq f(y)$. $f$ is said to be a Monotonic Function on the interval $[a, b]$ if $f$ either either increasing or decreasing on $[a, b]$.

For example, if $f : \mathbb{R} \to \mathbb{R}$ is defined to be $f(x) = x^2$ then $f$ is a decreasing function on any interval contained in $(-\infty, 0]$:

Furthermore, $f$ is an increasing function on any interval contained in $[0, \infty)$.

There are many important properties of monotonic functions. For example, if $f$ is an increasing function on the interval $[a, b]$ and $x, y, z \in [a, b]$ and such that $x < y < z$ and if $f(y^{-}) = \lim_{a \to y^{-}} f(a)$ and $f(y^+) = \lim_{a \to y^{+}} f(a)$ then the following inequality holds:

(1)

\begin{align} \quad f(x) \leq f(y^{-}) \leq f(y) \leq f(y^{+}) \leq f(z) \end{align}

The following graph illustrates the inequality above for increasing functions:

For decreasing functions and for $x < y < z$ we have that:

(2)

\begin{align} \quad f(x) \geq f(y^{-}) \geq f(y) \geq f(y^+) \geq f(z) \end{align}