New properties of all real functions

In a former papert the author communicated a number of properties of every real function f (x) ; these were stated in terms of the successive saltus functions associated with a given function. The present paper makes no use of the saltus functions, and the new properties are direct qualifications of f (x). Since f (x) is entirely unrestricted, except, of course, that it is defined t and therefore finite for every real x-these qualifications are consequences of nothing else than that f (x) is a function. A new light is thus thrown upon the nature of a function. The new properties are of two types, descriptive and metric; the former are concerned with density, and the latter with measure (Lebesgue). For the sake of greater concreteness of exposition, we shall discuss, for the most part, planar sets and real functions of two real variables.