This page is intended to be a part of the Real Analysis section of Math Online. Similar topics can also be found in the Calculus section of the site.

Functions Bounded on a Set

Definition: Let $f : A \to \mathbb{R}$ be a function. Then $f$ is said to be bounded on the set $A$ if there exists a real number $M \in \mathbb{R}$, $M > 0$ such that $\forall x \in A$ we have that $\mid f(x) \mid ≤ M$.

We should note that the definition of a function being bounded is analogous to the definition of a sequence being bounded. A function $f$ is bounded if its range is bounded, that is there exists values $x_1, x_2 \in A$ such that $f(x_1) ≤ f(x) ≤ f(x_2)$ for all $x \in A$.

For example, consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \sin x$. We know that the values of $\sin x$ are bounded below by $-1$ and bounded above by $1$. Therefore we can say that $\sin x$ is a bounded function on the set $\mathbb{R}$ since $\forall x \in \mathbb{R}$ we have that $\mid \sin x \mid ≤ 1$.

Definition: Let $f : A \to \mathbb{R}$ be a function. Then $f$ is said to be unbounded on the set $A$ if for any $M \in \mathbb{R}$, $M > 0$ there exists an $x_M \in A$ such that $\mid f(x_M) \mid > M$.

For example, consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^3$. We know that this function is not bounded. Given any positive real number $M$ no matter how large, we can find an $x_M \in \mathbb{R}$ such that $\mid f(x_M) \mid > M$.

To see this, let $M > 0$ be given. Then let $x_M = M$. Then $\mid f(x_M) \mid = \mid M^3 \mid > M$.

Another examples of a function that is not bounded is $f : \mathbb{R} (0, \infty) \to \mathbb{R}$ defined by $f(x) = \frac{1}{x}$. We can see this is unbounded since for any $M > 0$, we can choose $x_M = \frac{1}{M+1} > 0$. Therefore $\mid f(x_M) \mid = f(x_M) = \frac{1}{x_M} = M+1 > M$.