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.