Bounded Function¶

Definition¶

Let $X\subseteq \mathbb{R}\text{,}$ and let $f\colon X \to\mathbb{R}$ be a function. We say that $f$ is bounded from above if there exists a real number $M$ such that $f(x) \leq M$ for all $x \in X\text{.}$ We say that $f$ is bounded from below if there exists a real number $M$ such that $f(x) \geq M$ for all $x \in X\text{.}$ We say that $f$ is bounded if there exists a real number $M\geq 0$ such that $|f(x)| \leq M$ for all $x \in X\text{;}$ we say that $f$ is unbounded in the absence of such a real number $M\text{.}$

Interpretation¶

A bounded function is one whose values remain within fixed upper and lower bounds throughout its domain: its graph fits in a horizontal strip of finite width.

An oscillating function whose graph stays inside a shaded band between the horizontal lines at -M and M.

The graph oscillates, but it never leaves the shaded band between $-M$ and $M$ ; this is the condition $|f(x)|\le M$ for all $x$ . One number $M$ works for the whole domain. The function need not touch the edges of the band, and any larger $M$ would do as well.

The three notions in the definition fit together: $f$ is bounded if and only if it is bounded from above and from below, since $m\le f(x)\le U$ for all $x$ gives $|f(x)|\le\max\{|m|,|U|\}$ $max {∣ m ∣, ∣ U ∣}$ . Boundedness depends on the domain: $f(x)=\frac{1}{x}$ is unbounded on $(0,1]$ , but bounded on $[1,\infty)$ , where $0<f(x)\le1$ $1$ . In terms of sets, $f$ is bounded exactly when its range is a bounded set, and then the range has a supremum and an infimum.

A common slip is to expect a bounded function to reach its bounds. The function $f(x)=x$ on $(0,1)$ is bounded by $1$ , but no value equals $1$ and no value is the largest. Later, the extreme value theorem gives conditions under which a largest value is attained.

Key Results¶

If a function is bounded, then its range is a bounded set.
Continuous functions on closed and bounded intervals are bounded.
Boundedness is preserved under addition and multiplication by a constant.

Examples¶

$f(x)=\sin x$ is bounded on $\mathbb{R}\text{.}$
$f(x)=x$ is not bounded on $\mathbb{R}\text{.}$

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 4

Symbols: element of, infinity, real numbers, subset

Last updated: 20 October 2026 03:02