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On the Oscillation Functions derived from a Discontinuous Function

Published online by Cambridge University Press:  20 January 2009

L. R. Ford
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In this paper are introduced what we shall term “successive oscillation functions.” These functions are derived from functions of a real variable. The word “function” as here used has its widest meaning. We say y is a function of x in an interval of the the x-axis, if given any value of x, in the interval one or more values of y are thereby determined. The values of the function may be determined by any arbitrary law whatsoever. We shall deal with discontinuous functions; the theorems will be true for continuous functions, but will be trivial, except in the case of functions which are discontinuous and whose points of discontinuity are infinite in number. We shall assume in what follows that the values of the function lie between finite limits.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 33 , February 1914 , pp. 139 - 142
Copyright
Copyright © Edinburgh Mathematical Society 1914

References

* Baire, , Lecons sur les functions discontinues, Sec. 45, p. 70Google Scholar. Hobson [Theory of Functions of a Real Variable] uses the term oscillation somewhat differently. In forming M(f, a) and m(f, a) the values of the function at the points of the interval exclusive of the point a are considered. He uses the term saltus for oscillation as here defined.

† Baire, , loc. cit., Sec. 46, p. 73Google Scholar; Sec. 48, p. 77.

* Neither this theorem nor the following one is true for the oscillation as defined by Hobson.

† Here again there is a difference of definition. Harnack, [Math. Ann. 19 (1882), p. 242CrossRefGoogle Scholar, and 24 (1884), p. 218] adds the restriction that the points of discontinuity shall be of content 0. While Harnack's definition is useful in the theory of integration, it narrows the application of these functions in other fields. In the work of Baire, done since the publication of Harnack's papers, on the approach to discontinuous functions by continuous functions, the definition ia as we have given it.