A real- or complex-valued function is bounded when the moduli of all its values have one finite upper bound.
An unbounded function has no finite bound on the moduli of its values.
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A bounded function is a mathematical function that has a limited range of values. Specifically, a function \( f(x) \) is considered bounded if there exists a real number \( M \) such that for every input \( x \) in the domain of the function, the absolute value of the function output is less than or equal to \( M \).