Bounded function

A function is bounded if its values stay within some finite distance of zero, or equivalently lie in a bounded subset of its codomain when that codomain has a metric.

Bounded set: A subset of a metric space contained within some finite-radius ball. A function is bounded exactly when its image is a bounded set.
Bounded operator: A linear map between normed spaces whose output norm is bounded by a constant times the input norm. This specialized form of boundedness controls how a linear function scales every input.
Riemann integrability: Integrability defined by convergence of upper and lower sums over partitions. A bounded function on a closed interval is necessary for Riemann integrability.
Unbounded function: A function whose values have no finite upper bound in magnitude. It is the direct negation of boundedness, as with reciprocal behavior near zero.
Supremum: The least upper bound of a set in an ordered space. The supremum of absolute function values gives the sharpest bound when it exists.
Uniform continuity: Continuity with one input-distance threshold that works uniformly across the domain. A bounded derivative often provides a global estimate proving uniform continuity on an interval.
Lebesgue integrability: Integrability defined by measuring a function's values with respect to a measure. Boundedness helps guarantee integrability on finite-measure spaces, given measurability.
Continuous function: A function whose value changes continuously with its input at every point of its domain. Continuity alone does not ensure boundedness on noncompact domains.
Absolute value: The nonnegative magnitude of a real or complex number. Scalar-valued boundedness is expressed by bounding absolute values.
Extreme value theorem: A theorem stating that a continuous real-valued function on a compact space attains its maximum and minimum. Compactness and continuity guarantee boundedness and actual attainment of extreme values.

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