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.