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On accessibility of plane sets and differentiation of functions of two real variables

Published online by Cambridge University Press:  24 October 2008

R. O. Davies
R. O. Davies
Affiliation:
St Catharine's CollegeCambridge
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Extract

1. The paper is in three parts, of which the first is devoted to the proof of certain lemmas, which form the basis for the results proved in Parts II and III, and which are summed up in Lemma 6, §3. In Part II we consider questions relating to linear accessibility: a member of a set of points in the plane is said to be (linearly) accessible if through it there exists a straight line (infinite in both directions) containing no other point of the set. The main results are Theorems 3 and 5. Theorem 5 extends the result of Nikodym (2) that a set can be constructed which is of full measure in a square, and each point of which is accessible. Theorem 3 was conjectured by Besicovitch.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 2 , April 1952 , pp. 215 - 232
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

(1)Mossaheb, G. H.On differentiation and Denjoy-behaviour of functions of two real variables. Proc. Camb. phil. Soc. 46 (1950), 28–45.CrossRefGoogle Scholar
(2)Nikodym, O.Sur la mesure des ensembles plans dont tous les points sont rectilinéairement accessibles. Fundam. Math. 10 (1927), 116–68.Google Scholar