Bounded Function

analysis

Definition

Bounded Function

A real-valued function $f:D\to \mathbb{R}$ is called bounded if its image $f(D)$ is a bounded subset of $\mathbb{R}$.

This means there exist real constants $m$ and $M$ such that for all $x\in D$:

$$m\le f(x)\le M$$

Equivalently, $f$ is bounded if there exists a constant $K\ge 0$ such that $|f(x)|\le K$ for all $x\in D$. If such constants do not exist, the function is said to be unbounded.

Properties

• Operations: The sum, difference, and product of two bounded functions are also bounded functions.
• Continuity on Compact Sets: Any continuous function defined on a compact set (e.g., a closed and bounded interval $[a,b]$) is necessarily bounded (see Extreme Value Theorem).
• Supremum and Infimum: If a function is bounded, its supremum and infimum exist as finite real numbers.
• Relation to Limits: If ${\lim }_{x\to \infty }f(x)=L$(finite), then $f$ is bounded on some interval $[a,\infty )$.