Quantifying discontinuity

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.