Does Hahn-Banach yield a discontinuous additive $f : \mathbb{R} \to \mathbb{R}$? (open)

Work in ZF, without the axiom of choice, and write HB for the Hahn-Banach theorem.

Question. Does HB imply the existence of a discontinuous additive function $f : \mathbb{R} \to \mathbb{R}$, that is, a solution of Cauchy's functional equation $f(x + y) = f(x) + f(y)$ that is not of the form $f(x) = cx$?

Status: open. Like the companion question about a Hamel basis, this is recorded as open in Howard and Rubin's catalogue of consequences of the axiom of choice, as the implication from their form for Hahn-Banach to their form for a discontinuous additive function on the line.

Not proved in this library, and not proved anywhere. No page here depends on the existence of a pathological solution of Cauchy's equation, and this library develops neither functional analysis nor ZF model construction.

What is known, and what would settle it. For an additive $f : \mathbb{R} \to \mathbb{R}$, being continuous, being linear over $\mathbb{R}$, being measurable, being bounded on some set of positive measure and being bounded on some interval are all the same condition, so a discontinuous additive function is an extremely wild object: its graph is dense in the plane. The axiom of choice (The Axiom of Choice ↗) produces one immediately from a Hamel basis for $\mathbb{R}$ over $\mathbb{Q}$, by choosing a $\mathbb{Q}$-linear map that is not $\mathbb{R}$-linear. In the other direction, granted the consistency of ZF, ZF + DC cannot produce one: in Solovay's model, and in Shelah's 1984 strengthening that removes the inaccessible cardinal, every set of reals has the Baire property, and then every additive $f : \mathbb{R} \to \mathbb{R}$ is continuous. Shelah's version is what makes the consistency hypothesis just Con(ZF): Solovay's model on its own would need an inaccessible. So the statement sits strictly between ZF and AC, exactly as HB does, and the question is again how the two are ordered. Settling it means a ZF derivation from HB, or a model of ZF with HB and no discontinuous additive function.

Two nearby results sharpen what is at stake. Larson and Shelah (2026) build a model of ZF + DC with a discontinuous additive endomorphism of $\mathbb{R}$ but no Hamel basis for $\mathbb{R}$, so this consequence is strictly weaker than the Hamel basis in that setting and the two open questions are genuinely distinct. And the $\mathbb{R}$-linear analogue is at least as expensive: in the same ZF + DC model in which every set of reals has the Baire property, every linear functional on a Banach space is continuous, so the existence of a discontinuous linear functional on an infinite-dimensional Banach space is itself not provable in ZF + DC. It follows from the axiom of choice, but it is not known to this library's sources to be equivalent to it, and nothing here claims that it is. The function asked about above is only $\mathbb{Q}$-linear, which is what leaves room for it to be cheaper than either.

Why it matters here. A discontinuous additive function is the smallest and most-cited pathology in real analysis whose existence is not a theorem of ZF. Any statement of the form "the only additive functions are the linear ones" is a statement about the ambient set theory, not about the reals, and the library's choice ledger is where that has to be recorded. This item records that the exact price of the pathology, measured against Hahn-Banach, is unknown.