On the uniform approximation of Cauchy continuous functions (Q290605)

Let \((X,d)\) and \((Y,\rho)\) be metric spaces. A mapping \(f:X\to Y\) is called Cauchy continuous if it preserves Cauchy sequences (i.e., \((f(x_n))\) is Cauchy in \(Y\) for every Cauchy sequence \((x_n)\) in \(X\)), and Cauchy-Lipschitz if its restriction to any Cauchy sequence in \(X\) is Lipschitz.NEWLINENEWLINEThe authors present some properties of these classes of functions as well as their relations with other classes of functions -- Lipschitz, locally Lipschitz and Lipschitz in the small functions.NEWLINENEWLINEThe main result of the paper (Theorem 4.5) asserts that every Cauchy continuous function \(f: X\to Y\), where \((X,d)\) is a metric space and \((Y,\|\cdot\|)\) is a Banach space, can be uniformly approximated by Cauchy-Lipschitz functions.