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Monotonicity of Functions

Distinguish strict increase from nondecrease. Worked examples explain derivative tests, interval endpoints, and solving inequalities with monotonicity.

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Monotonicity of Functions

Monotonicity describes how a function's values change as its input increases. A precise statement identifies both the interval being considered and whether equal output values are allowed.

1. Strict and non-strict monotonicity

Let $I$ be an interval within the domain of $f$. For every pair $x_1,x_2\in I$ with $x_1<x_2$, use the following definitions:

Nondecreasing and nonincreasing functions may have equal values at different inputs. A constant function, for example, is both nondecreasing and nonincreasing, but it is not strictly monotonic on an interval containing distinct inputs.

Terminology varies between textbooks: “increasing” and “monotonically increasing” may mean either strictly increasing or nondecreasing. This tutorial distinguishes the two. The examples below use the strict meaning; when working with another source, check its definition.

A function can be monotonic on one interval without being monotonic on its entire domain. Always state the interval with your conclusion.

2. How to test for monotonicity

Using the derivative

Suppose $f$ is continuous on $I$ and differentiable in its interior. Check the derivative at every interior point:

• If $f'(x)>0$ throughout the interior, $f$ is strictly increasing on $I$.
• If $f'(x)<0$ throughout the interior, $f$ is strictly decreasing on $I$.
• If $f'(x)\ge0$ throughout the interior, $f$ is nondecreasing; if $f'(x)\le0$, it is nonincreasing.
• If $f'(x)=0$ throughout the interior, $f$ is constant on the interval.

A strictly increasing function need not have a positive derivative at every point. For example, $f(x)=x^3$ is strictly increasing on $\mathbb R$, even though $f'(0)=0$. A zero derivative at one point does not, by itself, rule out strict monotonicity.

Using the definition

Take arbitrary inputs $x_1,x_2\in I$ with $x_1<x_2$ and examine $f(x_1)-f(x_2)$. If this difference is negative for every such pair, the function is strictly increasing; if it is positive, the function is strictly decreasing. Replacing these strict inequalities with $\le$ or $\ge$ gives the tests for nondecreasing or nonincreasing functions.

Checking a few selected points is not enough to establish monotonicity on an entire interval.

3. Worked examples

Example 1: Find the intervals of increase and decrease

Find the intervals on which $f(x)=x^3-3x$ is increasing or decreasing.

The domain is $\mathbb R$, and

$f'(x)=3x^2-3=3(x-1)(x+1).$

• For $x<-1$ or $x>1$, the derivative is positive.
• For $-1<x<1$, the derivative is negative.
• At $x=\pm1$, the derivative is zero.

The function is strictly increasing on $(-\infty,-1]$ and $[1,+\infty)$, and strictly decreasing on $[-1,1]$. These endpoints may be included because the function is continuous there. Having a zero derivative at an endpoint does not prevent strict monotonicity on the interval.

Example 2: Solve an inequality using strict increase

Suppose $f$ is strictly increasing on $[0,+\infty)$ and $f(2)=5$. Solve $f(x)<5$ within that interval.

Replace $5$ with $f(2)$. Strict increase gives

$f(x)<f(2)\iff x<2.$

Combining this with $x\ge0$ gives the solution set $[0,2)$.

The word “strictly” matters. For a nondecreasing function, $f(x)<f(2)$ still implies $x<2$, but the reverse implication may fail: the function could remain equal to $5$ over an interval.

4. Conditions to check

• Start with the domain. Exclude zero denominators and check restrictions such as nonnegative radicands for real even roots.
• Check separate intervals separately. Being decreasing on two intervals does not establish decrease on their union. For instance, $1/x$ is strictly decreasing on each of $(-\infty,0)$ and $(0,+\infty)$, but $-1<1$ gives $f(-1)=-1<f(1)=1$. That pair violates decrease across the entire domain.
• Keep track of strict inequalities. Before replacing an inequality between function values with one between inputs, check which definition the problem uses.

Sources and review

Content reviewed: 12 September 2026.

Jiří Lebl, Basic Analysis, §3.6: monotone and strictly monotone functions