Definition:Locally Bounded/Family of Mappings

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Let $M = \struct {A, d}$ be a metric space.

Let $\FF = \family {f_i}_{i \mathop \in I}$ be a family of mappings defined on $M$.

Then $\FF$ is said to be locally bounded if and only if:

for all $x \in A$, there is some neighbourhood $N$ of $x$ such that $\FF$ is uniformly bounded on $N$.