In the first part of this paper the notion of natural metric on the set of natural numbers is defined. It is such metric that the completion of N is a compact metric space that a probability borel measure exists in order that the sequence { n} is uniformly distributed. A necessary and sufficient condition where a given metric is natural. Later we study the properties of sequences uniformly continuous with respect to given natural metric. Inter alia Theorems 5 and 8 characterise the continuity of distribution function
Categories: math.NT