Hello,

I want to show the following :

Let U be a open subset of IR^2.

If f is constant on each connected component, then f:U->\IR is locally constant.

My Idea was this:

Let x be a arbitrary point of U. We have to show that there is a (open) nbh. V of x, s.t.

f|V is constant.

We know x is an elm. of a conn. component V' of U and f is constant on V'.

Now i couldn't show that there exist an (open) nbh. V of x, s.t. V is a subset of V'.

Is this the right way? how can i show this property of conn. components.

Thank you for your help