Monotonicity

Monotonicity describes whether a function “rises” or “falls” consistently as $x$ increases.

Definition — Increasing, decreasing, monotonic function

A function $f$ is said, on a subset $I$ of its domain, to be:

• increasing if $x_1 < x_2 \implies f(x_1) < f(x_2)$ (that is, as $x$ increases $y$ increases);
• decreasing if $x_1 < x_2 \implies f(x_1) > f(x_2)$;
• monotonic if it is increasing or decreasing (strictly).

If the implication holds with $\le$ and $\ge$ instead of $<$ and $>$, we speak of a non-decreasing or non-increasing function.

Property — Strict monotonicity and injectivity

If $f$ is strictly monotonic on the whole of its domain $A$, then it is injective. The converse is not true: an injective function can oscillate. For example $f(x) = -x$ on $x<0$ and $f(x)=x$ on $x\ge 0$ is injective but not monotonic on the whole of $\mathbb{R}$.

Links

Topics: Functions and properties
Concepts: Injective function · Monotonicity