A null set that is the discontinuity set of no function whatsoever

There is $E \subseteq \mathbb{R}$ with $\lambda(E) = 0$ such that no function $g : \mathbb{R} \to \mathbb{R}$, Riemann integrable or not, has $E$ as its set of points of discontinuity.

The reason is a mismatch of descriptive complexity, not of size. For every $g : \mathbb{R} \to \mathbb{R}$ the set of points of discontinuity is an $F_{\sigma}$ set, a countable union of closed sets. So it suffices to exhibit a null set that is not $F_{\sigma}$: for instance a null set of the second Baire category cannot be $F_{\sigma}$, since a closed null set is nowhere dense and a countable union of nowhere dense sets is of the first category. A Lebesgue measurable set of measure zero that is not Borel serves as well.

This is the standard corrective to a careless reading of Lebesgue's criterion. That criterion says a bounded function on $[a,b]$ is Riemann integrable exactly when its discontinuity set is null; it does not say that every null set arises as a discontinuity set, and the example above shows that most do not.

Not proved in this library. It is recorded with a citation to Gelbaum and Olmsted and used in no proof here.

What would prove it. The $F_{\sigma}$ theorem for discontinuity sets is elementary and already in scope, since the set where the oscillation of $g$ is at least $1/n$ is closed. The remaining ingredient is a null set that is not $F_{\sigma}$, and the cheapest construction covers $\mathbb{Q}$ by open intervals of total length below $1/k$, intersects the resulting $G_{\delta}$ sets, and argues by Baire category; the Baire category theorem for complete metric spaces is in scope here, and so is the elementary notion of a null set. The alternative construction, through a measurable non-Borel set, does need the measure track (Lebesgue measure and the Lebesgue integral ‡).

Which page it serves. The Riemann integral page and the Cantor set, Baire and measure zero page, immediately after Lebesgue's criterion, and beside the counterexample that a Riemann integrable function may have a dense discontinuity set.

A candidate for undeferral. Like A bounded semicontinuous function equal almost everywhere to no Riemann integrable function ‡, this was listed among the measure-theoretic entries of Gelbaum and Olmsted's chapter 8, but the Baire route uses only machinery this library already intends to have. It should be re-examined when the Cantor set, Baire and measure zero page is authored.