On the Representation of Numbers in the Form

The points of E are now enclosed in the open intervals ĉ y, so that each point is inside of an infinite number of intervals, and (x) is defined to be the sum of all the a-intervals or parts thereof which lie to the left of x. Thus cj>(x) is monotone and can easily be shown to be absolutely continuous as follows: If i + j = N is chosen sufficiently large, the N(x) formed for this finite set of intervals will be absolutely continuous and as near as we please to {x) for all values of x. Hence (x) is absolutely continuous. If the set E is not an inner limiting set, the set E" = E + E, which lies inside an infinite number of a intervals, will be such a set, and (x) will have an infinite derivative at all the points of E" and no others. The set E may itself be an inner limiting set, in which case E = 0. It would be interesting to determine whether all absolutely continuous functions are of the form