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



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 //, it will be less 

 than /ul; let it equal /a— a, where or is some positive 

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

 of Sx such that \ Sy \ < 5-0-; 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 — Jo-; 



A Mj M, M :; M 4 M B 



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

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



