📂Functions

Monotonic Functions, Increasing Functions, Decreasing Functions

Definition

Let’s assume the function $f:[a,b] \rightarrow \mathbb{R}$ is given. For $x_{1}$, $x_{2}$, $\in [a,b]$

$$x_{1} \lt x_{2} \ \implies f(x_{1}) \le f(x_{2})$$

If it satisfies, then $f$ is said to be monotonically increasing or $f$ is called a monotone increasing function. Conversely,

$$x_{1} \lt x_{2} \ \implies f(x_{1}) \ge f(x_{2})$$

If it satisfies, then $f$ is said to be monotonically decreasing or $f$ is called a monotone decreasing function.

If $f$ is either a monotone increasing function or a monotone decreasing function, then $f$ is referred to as a monotone function.

Explanation

To monotonically increase means that as the variable increases, the function value does not decrease at least. Conversely, to monotonically decrease means it does not increase at least.

Definition

The below expression

$$x_{1} \lt x_{2} \implies f(x_{1}) \lt f(x_{2})$$

If satisfied by $f$, it is called a strictly increasing function. Conversely,

$$x_{1} \lt x_{2} \implies f(x_{1}) \gt f(x_{2})$$

If satisfied by $f$, it is called a strictly decreasing function.