Extension of Range of Functions

A well known and important theorem of analysis states that a function f(x) which is continuous on a bounded closed set E can be extended to the entire space, preserving its continuity. Let us consider a metric space S and a function f(x) defined and possessing a property P on a subset E of S. We shall for the sake of brevity say that ƒ (x) can be extended to S preserving property P , if there exists a function 4>(x), defined and possessing property P on all of S, which is equal to f(x) for all x on E. Our present object is to establish an easily proved theorem which both includes the classical theorem stated above, and also shows that functions satisfying a Lipschitz or Holder condition on an arbitrary set E can be extended to 5 preserving the Lipschitz or Holder condition. An advantage of the present procedure is that it yields an explicit formula for the extension, f