Unbounded

Unbounded is a mathematical term describing objects that are not contained within any finite bound. In real analysis, a subset S of the real numbers is unbounded if for every real number M there exists an element x in S with |x| > M. An object may be unbounded above, unbounded below, or both. Boundedness is the opposite notion; a set is bounded if there exists a finite bound B such that |x| ≤ B for all x in the set. In the extended real number system, unbounded above corresponds to a supremum of +∞.

Examples and consequences. The set of natural numbers is unbounded above. The sequence n, or the function f(x) = x^2, is unbounded on the real line. By contrast, the interval [−1, 1] is bounded, and any continuous function on it is bounded by the extreme value theorem. A function can be unbounded on a domain even if it is bounded on a compact subset; for instance, a function may grow without bound as its argument tends to infinity. When a function is unbounded above on a domain, its supremum in the extended real numbers is +∞; if it is unbounded below, its infimum is −∞.

Context and generalizations. In metric and normed spaces, boundedness is defined in terms of distances from a fixed point or in terms of a norm. Some spaces naturally generalize the idea of boundedness, while others require alternative notions. In broader usage, “unbounded” can describe quantities or processes that have no predetermined limit, such as unbounded growth in models or unbounded resources in theoretical discussions.

Unboundedness is a recurring consideration in theorems and proofs, often serving as a boundary condition for convergence, compactness, or integrability.