The Set of Universal Interpolating Functions is Nowhere Dense

TL;DR

This work establishes that the set of $X$-universal (interpolating) functions is topologically small in the space of real-valued continuous functions on $\mathbb{R}$ endowed with the metric $d_{\infty}$, provided $X$ satisfies $\mathbf{0}\in X$ and $\operatorname{span}(\mathbf{1})+X\subseteq X$. The main technique is a diagonal perturbation construction: for any $f\in C(\mathbb{R})$, one builds a sequence $f_n=f+h_n$ with $h_n=\sum_k h_{n,k}$ supported on disjoint blocks so that $f_n\notin U_{\text{N}}(X)$ while $d_{\infty}(f_n,f)=\|h_n\|_{\infty}=1/n\to 0$, and the complement of $U_{\text{N}}(X)$ is dense; coupled with the density of piecewise-affine functions, this yields that $U_{\text{N}}(X)$ is nowhere dense, hence $U(X)$ is nowhere dense as well. The paper also discusses extensions, open questions on prevalence, and corollaries for $X=\ell^{\infty}$, highlighting that algebraically large sets can be topologically small. Overall, the results clarify the topological rarity of universal interpolation phenomena in $C(\mathbb{R})$.

Abstract

In 1998, Benyamini introduced and proved the existence of universal interpolating functions. In the note we prove that the set of universal interpolating functions is nowhere dense in the space of continuous functions on $\mathbb{R}$. Several extensions and generalisations are also considered.

Theorems & Definitions (6)

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