Interpolation of sequences (Q851088)

Benyamini has shown that there is a continuous function \(f\colon \mathbb{R}\to \mathbb{R}\) such that for any sequence \((x_n)_{n\in\mathbb{Z}}\in [0,1]^{\mathbb{Z}}\), there is \(t\in \mathbb{R}\) such that \(x_n = f(t+n)\) for all \(n\in \mathbb{Z}\). Such a function is said to interpolate all sequences in \([0,1]^{\mathbb{Z}}\). In this note the authors extend Benyamini's result. More specifically, they show that there is a continuous function \(f\colon \mathbb{R}\setminus\mathbb{Q}\to \mathbb{R}\) that interpolates all sequences in \(\mathbb{R}^{\mathbb{Z}}\). This can also be done by a Baire class-1 function \(f\colon \mathbb{R}\to \mathbb{R}\).