Manchester Memoirs, Vol. xli. (1897), No. 10. 7 



In one class of cases, the truth of the theorem is indeed 

 obvious, viz., whenever the range considered admits of 

 being broken up into a finite number of intervals 

 within each of which the function steadily increases, 

 or steadily diminishes, as x increases. It appears, at 

 a later stage in the subject, that most mathematical 

 functions conform to this description; but the usual tests 

 by which we decide this question are based on reasoning 

 which presupposes the truth of the present theorem. 



It is therefore desirable, as a matter of logic, to have 

 a proof which shall postulate nothing as to the nature 

 of the function considered, except that it is continuous 

 according to the definition above given. 



In the geometrical representation, let 0A=a, 0B = b. 

 If at A the value of y is not equal to /x, it will be less 

 than jm; let it equal ju—er, where or is some positive 

 quantity. In virtue of the continuity, we can find values 

 of Sx such that \ Sy \ < icr; there will, therefore, be a 

 certain range extending to the right of A, but not reach- 

 ing to B, at every point of which y is not greater than 



A Mj M., M ;5 M, M B 



let M 1 mark the extremity of this range. Since y is 

 continuous, it is evident that at M x itself we shall have 



y = [A — ^cr. 



E 



