FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 145 



It follows that, if \i (0> X (0 can ^ c^ 08611 m 8UC ^ a wa y that 

 X( + 0) = li 



exists, then the bilateral limit of f(s) at the point a exists, and is equal to it. In 

 particular, the bilateral limit of f(s) exists at the point s, whenever 



(-*-0 



exists. 



14. In the preceding paragraphs we have developed the theory of upper and 

 lower bilateral limits in a form which is adapted to our immediate requirements, but, 

 on reviewing 11-13, the reader will find that, so far as concerns the definition of 

 these numbers and the fundamental inequality (33), f(s) may be any function which 

 has a Lebesgue integral in an interval (a, &), and s any point lying within this 

 interval. The definitions given are clearly applicable to the more general case, as 

 also are the relations (22) and (27). Choosing a fixed positive nuinlrer A such that 

 ^.s A, b^. .* + A, the reader will easily prove that 



" 



f 



Hence (28) may be adapted for the more general case by substituting 



in place of the left-hand member. Proceeding as alx>ve, we then obtain 



Y"[/(*-0+/(* + 0]*2s lim V*[/(-<)+/(-M)] *() (34) 



in place of (31). It now follows that the fundamental inequality (33) is valid at 

 each point a of the open interval (a, b). 



We may deduce from (33) an important property of functions which are integrable 

 in (a, 6) in accordance with the definition of LEBESGUE. For, from it, it will be clear 

 that ivhen 



istn at a point of the open interval (a, 6), it has a value independent of X i (0 

 Xi (t), namely, the bilateral limit of /(.<<) at s. 



It is not consonant with the plan of the present memoir to pursue this topic 

 further. 



1 5. We have now to consider the behaviour of 



JO 



VOL. CCXI. A. U 



