linguistic:paraphrasing
Related notions

Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded.

A bounded operator T: X → Y {\displaystyle T:X\rightarrow Y} is not a bounded function in the sense of this page's definition (unless T = 0 {\displaystyle T=0}), but has the weaker property of preserving boundedness; bounded sets M ⊆ X {\displaystyle M\subseteq X} are mapped to bounded sets T (M) ⊆ Y {\displaystyle T(M)\subseteq Y}. This definition can be extended to any function f: X → Y {\displaystyle f:X\rightarrow Y} if X {\displaystyle X} and Y {\displaystyle Y} allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.