# equicontinuous

## 1 Definition

Let $\mathcal{F}$ be a subset of $C(X,Y)$. A function $f\in\mathcal{F}$ is continuous at a point $x_{0}$ when given $\epsilon>0$ there is a neighbourhood $U$ of $x_{0}$ such that $d(f(x),f(x_{0}))<\epsilon$ for every $x\in U$. When the same neighbourhood $U$ can be chosen for all functions $f\in\mathcal{F}$, the family $\mathcal{F}$ is said to be equicontinuous. More precisely:

$\,$

Definition - Let $\mathcal{F}$ be a subset of $C(X,Y)$. The set of functions $\mathcal{F}$ is said to be equicontinuous at $x_{0}\in X$ if for every $\epsilon>0$ there is a neighbourhood $U$ of $x_{0}$ such that for every $x\in U$ and every $f\in\mathcal{F}$ we have

 $\displaystyle d(f(x),f(x_{0}))<\epsilon$

The set $\mathcal{F}$ is said to be equicontinuous if it is equicontinuous at every point $x\in X$.

## References

• 1 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
Title equicontinuous Equicontinuous 2013-03-22 18:38:10 2013-03-22 18:38:10 asteroid (17536) asteroid (17536) 6 asteroid (17536) Definition msc 54E35 msc 54C35 equicontinuity Equicontinuous2