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On differentiation and Denjoy-behaviour of functions of two real variables

Published online by Cambridge University Press:  24 October 2008

G. H. Mossaheb
G. H. Mossaheb
Affiliation:
167, Farhang StreetTeheranPersia
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Extract

The study of differential properties of functions of two real variables in the light of the modern theory of functions of real variables was started by H. Rademacher in 1919. Although much work has since been done on the subject, and a great many general results have been obtained, a number of questions have remained open, some of which will be discussed in the present work.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 1 , January 1950 , pp. 28 - 45
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

* Rademacher, H., Über partielle und totale Differenzierbarkeit, I. Math. Ann. 79 (1919), 340–59.CrossRefGoogle Scholar

† Stepanoff, W., Rec. Math. Soc. Math. Moscou, 32 (1925), 511–26.Google Scholar

‡ Ward, A. J., Proc. London Math. Soc. (2), 42 (1938), 266–73.Google Scholar

* Besicovitch, A. S., Math. Z. 41 (1936), 402–4.CrossRefGoogle Scholar

* Because the area of z = f(x, y) is finite.

† Besicovitch, A. S., Math. Z. 41 (1936), 402–4.CrossRefGoogle Scholar

* The reason for this restriction is that Ward's argument is based on the measurability of linear derivates of functions measurable (B) which follows from a general result, concerning functions measurable (B), due to Hausdorff, F. (Mengenlehre (1927), p. 274).Google Scholar

It is not known whether the linear derivates of functions which are merely measurable are measurable.

† Ward, A. J., Proc. London Math. Soc. (2), 42 (1936), 266–73.Google Scholar

‡ To fix ideas, p will be supposed horizontal, and the translations will be performed in the upward vertical direction.

* With a sufficiently close approximation.

* See the footnote to Theorem 5·1. The equality holds asymptotically.

† According to the notation described in § 4·5.

* As usual, MP is the perpendicular drawn from M to D n, and its length MP = d is supposed large compared with r 1. n (see § 6).

† In this, and similar expressions, PMP′ is to be taken as denoting the set of rays through M and inside the angle PMP′.

* β may be supposed to be such that this choice of α and δα is possible.

† For convenience, we put, for j = 1,

* Or Δ's or 's.

* Consequently all the squares about the various segments of all the cover a part of S of area less than

* See (7·1·2).

† See § 9.