In analysis there are certain theorems that tell under which conditions you can continuously extend a continuous functions to the closure/completion of its domain (which actually give the same set, since when talking about completions we have to be (at least) in a metric space).
The theorems I have in mind are these three:
- Every uniformly continuous mapping between a metric space and a complete metric can be extended uniformly continuous to a mapping between the completion of the domain and the range.
- Every continuous function on a dense subset can be continuously extend to the closure of that subspace (which is by definition of "dense", the whole space).
- The theorem from here.
What I'm looking for is a collection of theorems that tell me under which circumstances one can extend a continuous function beyond the completion or the closure of it's domain.
The only theorem of this kind that I know of is the famous Tietze-Urysohn extension theorem (which applies to function whose domain is already a closed set).
I'd be also happy with a reference, as long as the theorems listed there are quickly accessible (i.e. don't require reading through a thicket of abstract/very specialized definitions before getting to the theorems – the more "concrete" the theorems are (as in $\mathbb{R}^n$, or a metric space), the better.
Best Answer
Extensions problems come in two basic shapes:
So it makes sense that many extension theorems fall into one or the other class. Sometimes you just have to use two of them.
But extension theorems for Lipschitz functions, from the simple McShane-Whitney theorem to the more sophisticated Kirszbraun's theorem and its generalizations, do not care whether the domain is closed or not.
The recent two-volume book Methods of geometric analysis in extension and trace problems by Brudnyi and Brudnyi looks like the reference you want. It has separate chapters for results for $\mathbb R^n$ and for general metric spaces.