- #1

varygoode

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**[SOLVED] Uniform Continuity**

## Homework Statement

Let [tex] A \subset \mathbb{R}^n [/tex] and let [tex] f: A \mapsto \mathbb{R}^m [/tex] be uniformly continuous. Show that there exists a unique continuous function [tex] g: \bar{A} \mapsto \mathbb{R}^m [/tex] such that [tex] g(x)=f(x) \ \forall \ x \in A [/tex].

## Homework Equations

The definition for uniform continuity I am using is as follows:

Let [tex] f: A \subset \mathbb{R}^n \mapsto \mathbb{R}^m [/tex]. Then [tex] f [/tex] is uniformly continuous on [tex] A [/tex] if [tex] \forall \ \ \varepsilon > 0 \ \ \exists \ \ \delta > 0 \ \ s.t. \ \ \Vert f(x) \ - \ f(y) \Vert < \varepsilon \ \ \forall \ \ x, \ y \ \ \in A \ \ s.t. \ \ \Vert x - y \Vert < \delta [/tex].

## The Attempt at a Solution

I've got to admit, I think I'm pretty clueless about this one. I was thinking about some function

[tex] $ g(x)=\left\{\begin{array}{cc}x,&\mbox{ if }

x\in \bar{A} \backslash A\\f(x), & \mbox{ if } x\in A\end{array}\right $ [/tex].

x\in \bar{A} \backslash A\\f(x), & \mbox{ if } x\in A\end{array}\right $ [/tex].

But I'm not even sure if I'm allowed to do that or if it meets the criteria. Any help leading to a correct proof would be superb, thanks!