1.1. Let two real functions f and F be given such that f=f(x) is a continuous function for x [0, 1] and F== F(al, a2, * * *, a n, x) is a continuous function of n parameters and x [0, 1]. For simplicity of notation the point (a', a2, * * , an) in Euclidean n-space, En, is denoted by a. The domain of the parameters of F is denoted by P, a subset of En. The main problem in the theory of approximation of continuous functions may be stated as follows: Determine a*CP so that the deviation of the function F(a, x) from f(x) shall be minimized. Naturally one must define the deviation of F(a, x) from f(x) and different definitions lead to different theories. In this paper the deviation of F(a, x) from f(x) is taken to be maxxE[o,l] I F(a, x) -f(x) I. All maxima and minima are taken over xE [0, 1 ] unless otherwise stated. F(a*, x) is said to be a best approximation to f(x) if max I F(a*, x) -f(x) I _ max | F(a, x) -f(x) I for all aCEP. The results of this