DISCONTINUOUS FUNCTIONS. 243 



be of such a form that the signs of operation 

 will be sufficient to give the two values of Y. 

 This condition will be satisfied by the following 

 artifice : 



2Y = %) +f,(x) +|f(^)-f,(a;)l (1) 



where + is to be taken when x is less than a, 

 and - when j; is greater than a. Therefore, 

 the above equation may be written in the following 

 manner : 



Y = f(r) + f,(r) + <!.,(«, x) 5 f(^) - f(^) I 



(2). 



where *i(«, .v) must be such a function of (a, x) 

 as will satisfy the conditions, that when .v is less 

 than a we must have <J>, (a, x) — I. and when 

 .1' is greater than a, we must have $, (a, a;) 



In searching through the wide field of func- 

 tional analysis, various functions of (a, .v) may 

 possibly be found, that will satisfy the conditions 

 which the above considerations have imposed 

 upon them ; none more simple, perhaps, than 

 the following : 



