preprint · ArXiv.org · 2025

Quantifying discontinuity

Henry Adams, Florian Frick, M. Harrison, Nikola Sadovek, Matt Superdock

Cited by 0Open access

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.

Cite this paper

APA 7

Adams, H., Frick, F., Harrison, M., Sadovek, N., & Superdock, M. (2025). Quantifying discontinuity. ArXiv.org. https://doi.org/10.48550/arxiv.2511.07636

MLA 9

Adams, Henry, et al. “Quantifying discontinuity.” ArXiv.org, 2025, https://doi.org/10.48550/arxiv.2511.07636.

Chicago

Adams, Henry, Florian Frick, M. Harrison, Nikola Sadovek, and Matt Superdock. 2025. “Quantifying discontinuity.” ArXiv.org. https://doi.org/10.48550/arxiv.2511.07636.

Harvard

Adams, H. et al. (2025) ‘Quantifying discontinuity’, ArXiv.org. Available at: https://doi.org/10.48550/arxiv.2511.07636.

BibTeX

@misc{adams2025quantifying,
  title = {{Quantifying discontinuity}},
  author = {Adams, Henry and Frick, Florian and Harrison, M. and Sadovek, Nikola and Superdock, Matt},
  journal = {ArXiv.org},
  year = {2025},
  doi = {10.48550/arxiv.2511.07636}
}

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