Definition:Locally Bounded
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Definition
Locally Bounded Mapping
Let $M = \struct {A, d}$ be a metric space.
Let $f$ be a mapping defined on $M$.
Then $f$ is said to be locally bounded if and only if:
- for all $x \in A$, there is some neighbourhood $N$ of $x$ such that $f \sqbrk N$ is bounded.
Locally Bounded Family of Mappings
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 $N$.
Also see
- Results about local boundedness can be found here.