Geodesic
On C∞ Functions Analytic on Sets of Small Measure
Canadian mathematical bulletin, Tome 12 (1969) no. 1, pp. 25-30

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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. 
May, L.E. On C∞ Functions Analytic on Sets of Small Measure. Canadian mathematical bulletin, Tome 12 (1969) no. 1, pp. 25-30. doi: 10.4153/CMB-1969-003-5
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     title = {On {C\ensuremath{\infty}} {Functions} {Analytic} on {Sets} of {Small} {Measure}},
     journal = {Canadian mathematical bulletin},
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     year = {1969},
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     doi = {10.4153/CMB-1969-003-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1969-003-5/}
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