Tangential properties of sets and arcs of infinite linear measure

Abstract

The theorem on differentiability of a monotone function or, what is equivalent, on the existence of the tangent at all points of a rectifiable curve, is one of the most remarkable theorems of mathematics. I t is remarkable because it deals with the simplest and yet fundamental notions of analysis. The theorem is deep and it had never been guessed before. I t is also remarkable because it represented a fundamental tool in the rebuilding of analysis in the twentieth century. How has it happened that the theorem had not been discovered before? The problem on differentiability occupied mathematicians for something like a century before, first in attempts to prove differentiability of continuous functions and later in establishing cases of nondifferentiability. Among those who had been working we find the names of Cauchy, Riemann, Weierstrass. How is it that monotonicy had never come into consideration? But perhaps it came, and without methods of measure the problem was unsolvable? This is not true: the theory of measure is not needed for the proof. The simplest proof is not based on the theory of measure: it is proved that the set of points of a rectifiable curve at which no tangent exists can be included in a set of arcs of the curve, of arbitrarily small total length, and the set is arrived at without any use of theory of measure. The theory of measure was not needed for the proof, but the problem could not be set without ideas of measure. Riemann gave an example of a monotone function (indefinite integral) for which both the set of points of differentiability and the set of points of nondifferentiability are everywhere dense. In those days the idea that one of these sets is almost the whole of the interval and the other almost empty could not arise. As soon as Lebesgue developed the theory of measure, the theorem came as a most beautiful boon of new methods. On the basis of L-measure the study of differentiability of continuous functions reached a completeness in the theorems of Denjoy, in the second decade of this century. In the same decade Caratheodory introduced methods of linear measure in planes and spaces of higher dimensions. Ideas of Caratheodory were developed by