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 \( M \).
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A real- or complex-valued function is bounded when one finite constant controls all its values:For example, is bounded by one on the real line. The reciprocal is bounded by one on , but is an unbounded function on , showing that the domain of a function matters. Each value at an allowed input may be finite even when no common bound exists.