Approximations of continuous functions by squares

Abstract

Let C denote the space of continuous functions mapping [0,1] into itself and endowed with the sup metric. It has been shown that C² = {f ∘ f: ∈ C} is an analytic but non-Borel subset of C. This implies that there is no simple geometric characterization for a function being a square. In this paper we consider the problem of characterizing those functions which can be approximated by squares. In the first section we prove that any continuous function mapping a closed proper subset of [0,1 ] into [0,1 ] can be extended to a square. In particular this shows that C² is L_p dense in C. On the other hand, C² does not contain a ball when C is endowed with the sup metric. In the second section we prove that no strictly decreasing function can be uniformly approximated by squares, although the distance between the class of strictly decreasing functions and C² is zero. In the last section we investigate the function f(x) = 1 − x and show that for every g ∈ C and that ¼ cannot be improved.