On some singular monotonic functions which are strictly increasing

Abstract

almost everywhere, may be constant in every interval contiguous to a perfect set of measure zero: it is usually said, in this case, that f(x) is of the Cantor type. There are, however, monotonic continuous functions, purely singular, which are increasing in the strict sense, that is, f(x') >f(x) whenever x' >x. While the existence of functions of the Cantor type is almost intuitive and their construction is immediate by successive approximations, the existence of strictly increasing singular functions lies deeper. Actually, if we except Minkowski's function ?(x), of which we shall speak later (and whose singularity is by no means obvious); no simple direct construction of such functions seems to be known. Functions of this type usually have been obtained by convolutions of functions of the Cantor type and the proof that they are singular strictly increasing functions is somewhat difficult('). Thus, it seems to be of interest to give simple direct constructions of strictly increasing singular functions. 2. Let us consider, in the plane, the straight line PQ joining the point P of cartesian coordinates x, y, to the point Q of cartesian coordinates x+Ax, y+Ay, Ax>O, Ay>O. Let X0, Xi be two numbers, essentially positive, such that Xo+Xi = 1 (Xo0#1). Let us now consider the point R whose coordinates are