Boundedness

A subset of a metric space has boundedness when it lies in some ball of finite radius, equivalently when its diameter is finite. A function is bounded when its range is a bounded set. For real-valued functions, this means that $|f|$ has a finite upper bound.

Failure of boundedness: for every finite radius, some point lies outside the proposed bounding ball. A real-valued function is unbounded if its range is unbounded.