Pointwise Products of Uniformly Continuous Functions on Sets in the Real Line

Abstract

Even though the pointwise product of two continuous real-valued functions is contin uous, the pointwise product of two uniformly continuous real-valued functions need not be uniformly continuous. For example, the identity function on the real line R1 is uniformly continuous but its pointwise product with itself is not uniformly continuous. In [2] and [4] various sufficient conditions are given on uniformly continuous real valued functions in order that their pointwise product be uniformly continuous. The conditions are somewhat technical and often place additional assumptions on the func tions themselves; sometimes the conditions involve particular functions (for example, when one function is the identity function). The following problem is neither stated nor addressed: Characterize those sets in the real line on which the pointwise product of any two uniformly continuous real-valued functions is uniformly continuous. In other words, determine the sets in the real line on which all the real-valued uniformly con tinuous functions form a ring (under pointwise addition and pointwise multiplication). It is surprising to us that this problem has not been explicitly investigated, especially in view of the vast literature on rings and semigroups of continuous functions. Our purpose is to present a simple solution to the problem. After we prove our theorem, we comment on the general problem of characterizing those metric spaces on which all real-valued uniformly continuous functions are closed under pointwise multiplication. Our theorem is related in a satisfying way to a characterization of uniformly contin uous sets due to Levine [3]. We state Levine's theorem after we give two definitions. A set in R1 is called a uniformly continuous set provided that all continuous real valued functions on the set are uniformly continuous [3]. A set X in R1 is uniformly isolated provided that there is an c > 0 such that \x ? > c holds for all distinct points x and y in X. These definitions generalize in a straightforward way to metric spaces. Levine's theorem reads [3]: