Definition:Locally Bounded/Family of Mappings

Definition
Let $M = \left({X, d}\right)$ be a metric space.

Let $\mathcal F = \left \langle{f_i}\right \rangle_{i \mathop \in I}$ be a family of functions defined on $M$.

Then $\mathcal F$ is said to be locally bounded if for all $x \in X$, there is some neighbourhood $A$ of $x$ such that $\mathcal F$ is  uniformly bounded on $A$.