Unboundedness

Unboundedness is the property of a mathematical object not being contained within any bounded region. In real analysis and metric spaces, boundedness is defined relative to a metric or norm: a subset S is bounded if there exists a point x and a radius R such that S lies within the ball B(x, R). If no such ball exists, S is unbounded. The same idea applies to functions: a function f on a domain D is bounded if its range is contained in a bounded interval; otherwise f is unbounded. A sequence (a_n) is bounded if the set {a_n : n ∈ N} is bounded; if not, the sequence is unbounded. A set can be bounded above, bounded below, or bounded in both directions; if neither holds, it is unbounded.

Examples illustrate unboundedness. The set of natural numbers is unbounded above. The real line is unbounded in both directions. The function f(x) = x is unbounded above and below on R, and the function f(x) = e^x is unbounded above on R (though it is bounded below by 0). By contrast, the function sin(x) is bounded since its values lie between -1 and 1.

In analysis, unboundedness often relates to limits and divergence. A function may fail to have a finite limit at infinity or near a point because it grows without bound. A sequence that does not stay within any fixed range diverges. In functional analysis, the term also appears in the concept of unbounded operators, which are linear operators not bounded in the operator norm.