Entropy and the approximation of continuous functions

Abstract

The rate of decrease of co(f; 3) for ~ --, + 0 is a measure for the degree of continuity of the function f. In the following we shall relate the degree of continuity of a function f ~ C(X) to the degree of compactness of the underlying metric space (X, d) by inserting the sequence e, (X) of entropy numbers for the argument ~ in the modulus of continuity. The n-th entropy number ~, (X) is defined to be the infimum of all ~ > 0 such that there are open balls U(tl;e)={teX:d(t , tl)<e}, i <-iNn, of radius ~ which cover X, that is to say