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On C Functions Analytic on Sets of Small Measure

Published online by Cambridge University Press:  20 November 2018

L.E. May*
Affiliation:
North Carolina State University at Raleigh
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The original motivation for this work was the problem of determining whether the signum function of a real valued continuous function defined on the real line is Riemann integrable. This problem is considered in § 2 where an example of an infinitely differentiable function is presented which possesses a non-Riemann integrable signum function. Moreover, it is shown that, for any ∈ > 0, it is possible to construct such an example for which the set of points of analyticity has Lebesgue measure which is less than ∈. This appears to be a more interesting property than the one originally sought.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1969