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arXiv · 2511.07636

Quantifying discontinuity

Henry Adams·Florian Frick·Michael Harrison·Nikola Sadovek·Matt Superdock
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Abstract

Given a compact space $X$ that does not admit an embedding (an injective continuous function) into $\mathbb{R}^d$, we study the ''degree'' of discontinuity that any injective function $X \to \mathbb{R}^d$ must have. To this end, we define a scale invariant modulus of discontinuity and obtain general lower bounds, thus obtaining quantified nonembeddability results of Haefliger--Weber type. Moreover, we establish analogous lower bounds for simplicial complexes that do not admit an almost $r$-embedding in $\mathbb{R}^d$, thus obtaining a quantified version of the topological Tverberg theorem.

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Henry Adams, Florian Frick, Michael Harrison, Nikola Sadovek, Matt Superdock. 2025-11-10. Quantifying discontinuity. https://arxiv.org/abs/2511.07636

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