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On continuous functions with graphs of σ-finite linear measure

Published online by Cambridge University Press:  24 October 2008

James Foran
James Foran
Affiliation:
University of Wisconsin, Milwaukee, U.S.A.
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Extract

In this paper comparisons are made between the class of continuous functions of generalized bounded variation and the class of continuous functions with graphs having σ-finite length i.e. linear measure. An investigationof the differentiability and approximate differentiability of such functions discloses the fact that the latter class is considerably more extensivethan the former one. The following definitions will be needed:

(1) A function f is said to be of bounded variation (VB) on a set E if

where the supremum is taken over all sequences {[ai, bi]} of non-overlapping intervals with endpoints in E.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 1 , July 1974 , pp. 33 - 43
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

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