The Set of Universal Interpolating Functions is Nowhere Dense

TL;DR

This paper proves that the set of universal interpolating functions, introduced in 1998, is nowhere dense in the space of continuous functions on the real line, with various extensions and generalizations explored.

Contribution

It establishes that the set of universal interpolating functions is nowhere dense, providing new insights into their topological properties within continuous functions.

Findings

Universal interpolating functions form a nowhere dense set

Extensions and generalizations of the main result are discussed

The result clarifies the topological size of universal interpolating functions

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.