THE APPROXIMATION OF REAL FUNCTIONS IN THE HAUSDORFF METRIC

1. Preliminaries. Let Y be a metric space with metric d and for each y in Y let BX[Y] denote the closed ball of radius X about y. Following Valentine [14] if K C Y and X is positive, we call the set Bx[K] = Uy•KBx[y] the X-parallel body of K. If C and K are nonempty subsets of Y and there exists X> 0 such that Bx[C] D K and Bx[K] D C, then the Hausdorffdistance of C from K is given by D(C,K) = inf{X: Bx[C] D K and Bx[K] D C}. If no such X exists, then we let D(C,K) be infinity. If we identify sets with the same closure, then D is well defined on the ec•uivalence classes so determined. Moreover, D defines an extended real valued metric on CL(Y), the class of nonempty closed subsets of Y, called the Hausdorœf metric. A further discussion of this metric can be found in Berge [4], Castaing and Valadier [6], Kuratowski [9], or Nadler [ 11]. Now let X be a metric space with metric d and let C(X) denote the set of continuous real valued functions on X. The uniform distance between two functions f and gin C(X) is of course given bydl(f,g) = xS•lf(x)- g(x)l. Let us metrize X X R in a manner compatible with the product uniformity. For definiteness and computational simplicity we shall use the metric p defined by p[(Xl,O•l),(x2,o•2)] = max {d(Xl,X2),[O•l-'o•2 [}. Using this metric we can restrict the Hausdorff metric D defined on the closed subsets of X X R to the graphs of functions in C(X) to obtain a different metric d 2 on C(X). Explicitly, if f and g are in C(X) let us represent their graphs by'•and'•, respectively. Then d2(f,g) is defined by the formula d2(f,g) = inf{X: Bx[f] D g and Bx[g] D f} where the parallel bodies are taken with respect to the metric p. Clearly, dl(f,g) •&gt