Definition
Bounded Function
A real-valued function
is called bounded if its image is a bounded subset of . This means there exist real constants
and such that for all : Equivalently,
is bounded if there exists a constant such that for all . 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
) 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
(finite), then is bounded on some interval .