14: Proceedings of the Roijal Irish Academij. 



Suppose a set of open intervals {a J in tiie fundamental segment <.r < 1 

 be defined as follows : — Let oi have its centre at the point x = h Let a^i 

 and 022 be two equal intervals occupying the central parts of the remaining 

 segments. Let intervals be similarly constructed in the four remaining 

 segments and so on. We can choose these intervals so that their content 

 is 1 - *S', where ;S' is any positive number < 1. 



Xow, suppose that each interval a is divided into an enumerable set of 

 intervals constructed in the same manner as the original set. The whole 

 line is now divided into a set of intervals (aW), of content (1 - ^)(1 - Si). 

 Let each interval a'^' be again divided, and let this process be indefinitely 

 continued. The sequence of sets of intervals { a ) , { a'^) j , . . . &c. , define an inner 

 limiting set E of content 



Now, suppose that the numbers S, Si . . . &c., have been chosen so that 

 this product converges to a positive limit J. Then the content of E = J. 

 Let a function be defined as follows for all the points <.x ^ 1 : — 



f{x) = X + 1 for the points C [a] complementary to the set of 



intervals [a], 

 f{x) = X + i for those points of G [aOJ which are not contained 

 in C [a\, 



-, , 1 for those points of (7 (a'"'! which are not contained 



•'^ ' 2" in C I «("-')), 



and /'(.(•) = X for the points E. 



The function is discontinuous at the points 6'{o'"'} for all values of n. 

 Moreover, the saltus at all the points of {a'"' j is < — • Hence the set E consists 

 of the points of continuity of the function. 



Now, at all points in {a^"'), /'OO ^ ■*-' + h,; J 



where J„ = content of {a'">}, and 



Hence L=j.=^'^- 



