FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 127 



where is the upper limit of x( s t ) ln tne rectangle a, s t s 6,, a, s * : 6,. It 

 follows from the remark made above that 



. . ,.,.(12) 



at each point .s of (a a , b 2 ). 



Again, the second term on the right of (11) is not greater than 



and the integral just written tends to zero with rj, by a theorem due to LEBESGUE.* 

 Thus we have 



lim fcfc tmtf(t + + v)7-t+ dt = 



at each point * of (a a , b a ). Taking this in conjunction with (12), we see from (11) 

 that 



4L 



'. t)f(t + s)Jdt 



is a continuous function of * in (ci a , b a ). It may be proved in the same way that each 

 of the terms 



is continuous in this interval. 



It has now been shown that the positive series on the left of (10) has terms which 

 are continuous in (a a , b 2 ), and a sum function which is also continuous in this interval. 

 It follows from DINI'S theorem! that the series is uniformly convergent in (a,, b 3 ) ; 

 and hence that, as n tends to oo, 



converges to zero uniformly for a a S .s- ^ 6j. 



9.' Let us next suppose f(s) to be an unlimited function. We may show that 

 (13) converges uniformly to zero, provided that the normal functions />(*) satisfy the 

 requirement of 6. For, let us define a limited function /, (*) by the rule 



-0 !/()!> N; 



* Vide 'Le9ons sur les Series Trigonom&riques ' (1906), pp. 15, 16. 



t DIM, 'Fondamente per la teoria delle fuiizioni di variabili reali ' (Pisa, 1878), 99. See also YOUNG, 

 " On Monotone Sequences of Continuous Functions," 'Proc. Camb. Phil. Soc.,' vol. xiv., pp. 620-523. A 

 proof of the theorem will be found in HOBSON'S ' Theory of Functions of a Real Variable,' pp. 478-479. 



