# Definition:Locally Bounded

## Definition

### Locally Bounded Mapping

Let $M = \left({A, d}\right)$ 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 $f \left[{N}\right]$ is bounded.

### Locally Bounded Family of Mappings

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

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

Then $\mathcal F$ is said to be **locally bounded** if and only if:

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