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On the derivation of discontinuous functions
Published online by Cambridge University Press: 24 October 2008
Extract
Let f(x) be a real function of the real variable x, let P be any point lying on the graph of f(x) and let l be a ray from P making an angle θ (− π < θ ≤ π) with the positive direction of the x-axis. We say that θ is a derivate direction of f(x) at the point P if the ray l meets the graph of f(x) in a set of points having a limit point at P.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 35 , Issue 3 , July 1939 , pp. 373 - 381
- Copyright
- Copyright © Cambridge Philosophical Society 1939
References
† In the cases we have to consider this set of points is always measurable.
† Our segments are always enumerated from the left.
† Compare the set discussed in Chapter III of my paper, “Study of extreme cases with respect to the densities of irregular linearly measurable plane sets of points”, Math. Ann. 116 (1939), 371–3.Google Scholar
‡ Besicovitch, A. S., “On the fundamental geometrical properties of linearly measurable plane sets of points (II)”, Math. Ann. 115 (1938), § 10, p. 311, theorem 9.CrossRefGoogle Scholar
† The choice of n 0 depends, of course, upon the particular rectangle γN under consideration.
‡ The figure is drawn for the case when the point P lies to the right of and above our rectangle γnj. For the sake of clarity the segment XY has been omitted.
† Similar arguments arise at later stages in the proof but we do not stress them.