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Note on a Property of Continuous Arcs

Published online by Cambridge University Press:  24 October 2008

W. L. Ayres
W. L. Ayres
Affiliation:
University of Michigan
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Extract

1. In a note in these Proceedings, Mr S. Verblunsky proves the following result:

Theorem A. Let C be a continuous arc joining the points (x1, y0) and (x2, y0), where x2 > x1. Suppose that C does not cross the line y = y0. Then, given any positive h < x2 − x1, there is α yh such that some two points (ξ1, yh), (ξ2, yh), in which y = yh cuts C, satisfy the relation ξ2 − ξ1 = h.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 27 , Issue 4 , October 1931 , pp. 543 - 545
Copyright
Copyright © Cambridge Philosophical Society 1931

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References

* Proc. Camb. Phil. Soc. 26 (1930), 31.Google Scholar

* By the segment pq is meant the set of points between p and q, excluding the points p and q themselves.

† Mullikin, Anna M., ‘Certain theorems relating to plane connected point sets’, Trans. Amer. Math. Soc. vol. 24 (1922), 157, Theorem 2.CrossRefGoogle Scholar