A Function

Lingual

1. What something does or is used for.
   • Synonyms: aim, intention, purpose, role, use
2. A professional or official position.
   • Synonyms: occupation, office, part, role
3. An official or social occasion.
   • Synonyms: affair, occasion, social occasion, social function
4. A relation where one thing is dependent on another for its existence, value, or significance.
5. (mathematics) A relation in which each element of the domain is associated with exactly one element of the codomain.
   • Synonyms: map, mapping, mathematical function, operator, transformation
   • Hypernym: relation
6. (computing) A routine that receives zero or more arguments and may return a result.
   • Synonyms: procedure, routine, subprogram, subroutine, func, funct
7. (biology) The physiological activity of an organ or body part.
8. (chemistry) The characteristic behavior of a chemical compound.
9. (anthropology) The role of a social practice in the continued existence of the group.

Mathematics

Calculus

bounded function

A bounded function is a function whose values do not grow arbitrarily large in either the positive or negative direction. In other words, its entire output (range) is trapped between two finite real numbers.

Formal Definition

A real-valued function \(f: X \to \mathbb{R}\) defined on a set \(X\) is bounded if there exists a real number \(M \ge 0\) such that:

\[\vert{}f(x)\vert{} \le M \quad \text{for all } x \in X\]

Equivalently, \(f\) must be both:

1. Bounded above: There exists a real number \(U\) (an upper bound) such that \(f(x) \le U\) for all \(x \in X\).
2. Bounded below: There exists a real number \(L\) (a lower bound) such that \(f(x) \ge L\) for all \(x \in X\).

When both exist, every output satisfies \(L \le f(x) \le U\). Setting \(M = \max(\vert{}L\vert{}, \vert{}U\vert{})\) yields the absolute value condition.

Key Properties & Theorems

• Extreme Value Theorem: If a function \(f\) is continuous on a closed, bounded interval \([a, b]\), then \(f\) is guaranteed to be bounded and attains its maximum and minimum values.
• Algebra of Bounded Functions: If \(f\) and \(g\) are bounded on the same set \(X\), then \(f + g\), \(f - g\), and \(f \cdot g\) are also bounded.
• Supremum and Infimum: For any bounded real function, the set of outputs \(\{f(x) \mid x \in X\}\) has a well-defined supremum (least upper bound) and infimum (greatest lower bound).