This page is about Bounded Mapping in the context of Metric Space. For other uses, see Bounded.
Let $M$ be a metric space.
Let $f: X \to M$ be a mapping from any set $X$ into $M$.
Then $f$ is a bounded mapping if and only if $f \sqbrk X$ is bounded in $M$.
From Real Number Line is Metric Space, we can in theory consider defining boundedness on a real-valued function in terms of boundedness of a mapping into a metric space.
However, as a metric space is itself defined in terms of a real-valued function in the first place, this concept can be criticised as being a circular definition.