# Extension of a continuous function

1. Sep 18, 2010

1. The problem statement, all variables and given/known data

Let f : A --> Y be a continuous function, where A is a subset of X and Y is Haussdorf. Show that, if f can be extended to a continuous function g : Cl(A) --> Y, then g is uniquely determined by f.

3. The attempt at a solution

I think I can solve this on my own, but I got a bit stuck, perhaps I didn't understand the problem right, so let me check.

I assume an extension of a function in this case means that for any x in A, f(x) = g(x), and further on, I assume that g is uniquely determined by f if, for any x in Cl(A) , f(x) = g(x), too. Am I right on this one?

2. Sep 18, 2010

### Office_Shredder

Staff Emeritus
f(x) doesn't exist for points in Cl(A) necessarily. When they say that g is uniquely determined, it means that suppose g(x) and h(x) are continuous functions on Cl(A) such that f(x)=g(x)=h(x) on A. Then g(x)=h(x) for all points in Cl(A)

3. Sep 19, 2010

OK, thanks. I'll try to do it right away.

4. Sep 19, 2010