Extending a uniformly continuous function from a dense subset

[robertdeniro]

Homework Statement

we have 2 metric spaces (X, d) and (Y, d')

given:
1) A is a dense subset of X
2) Y is complete
3) there is a uniformly continuous function f: A->Y

let g: X->Y be the extension of f
that is, g(x)=f(x), for all x in A

is g uniformly continuous?

Homework Equations

The Attempt at a Solution

not sure where to start...

 

[micromass]

You could start by proving that such an extension exists and is unique. How would you define g?