Bounded function

Set

 context $\langle X,d\rangle$ … metric space
 definiendum $f\in \text{BoundedFunction}(X)$
 range $x,a\in X$ range $M\in \mathbb R$
 postulate $\exists a, M.\ \forall x.\ d(f(x),a) < M$

Discussion

For linear operators between normed vector spaces, being bounded for each input element and continuous coincide.

Reference

Wikipedia: Bounded function