Hilbert 13: Are there any genuine continuous multivariate real-valued functions?

Abstract:

This article begins with a provocative question: Are there any genuine continuous multivariate real-valued functions? This may seem to be a silly question, but it is in essence what David Hilbert asked as one of the 23 problems he posed at the second International Congress of Mathematicians, held in Paris in 1900. These problems guided a large portion of the research in mathematics of the 20th century. Hilbert’s 13th problem conjectured that there exists a continuous function $f:\mathbb {I}^3\to \mathbb {R}$, where ${\mathbb {I}=[0,1]}$, which cannot be expressed in terms of composition and addition of continuous functions from $\mathbb {R}^2 \to \mathbb {R}$, that is, as composition and addition of continuous real-valued functions of two variables. It took over 50 years to prove that Hilbert’s conjecture is false. This article discusses the solution.