Tietze and Hausdorff extension formulas

Given a continuous bounded real-valued function f defined on a closed subset A of a metric space X, we would like to extend it to a continuous function F on the entire space X. The first thought is that F(x) should be equal to the value of f at the nearest point of A. But of course such point may fail to exist, or to be unique, or to depend continuously on x. But the idea can be rescued.

Let d(x,x') denote the distance between two points on X. The distance from point x to set A is d(x,A)=\inf_{a\in A} d(x,a). This distance is strictly positive when x\notin A, although the infimum is not necessarily attained.

Tietze (1915, paper by subscription) proved the existence of a continuous extension by setting \displaystyle F(x)=\sup_{a\in A}\left(\frac{f(a)}{1+d(x,a)^2}\right)^{1/d(x,A)} (under the assumption \inf_A f>0, which can be achieved without loss of generality). The idea is that even though the supremum allows a to be any point of A, the “far-away” points have d(x,a)\gg d(x,A) which makes denominator large and thus prevents such points from affecting the supremum. In contrast, when a is the nearest or almost-nearest point, we have d(x,a)\approx 1/d(x,A) and as the distance tends to zero, the denominator approaches 1.

A few years later Hausdorff (1919, paper by subscription) offered a simpler and more natural formula: \displaystyle F(x)=\inf_{a\in A} \left(f(a)+\frac{d(x,a)}{d(x,A)}-1\right). The penalty is now assessed via addition rather than division, which eliminates the need to make f positive prior to extension.

The simplest nontrivial example is extension from a two-point set. I took X=\mathbb R^2; the set A consists of the points (0,0), (1,0), at which f is defined to be 0 and 1, respectively. Here is the extension F computed according to Hausdorff (not to scale):

Hausdorff formula

And this is its slice long the x-axis, true scale:

Slice

As you can see, the extension isn’t the best one could have in terms of continuity, but its attractive feature is that one does not need to compute anything like the modulus of continuity of f (or its Lipschitz constant, as with the McShane-Whitney extension operator).

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