Manchester Memoirs, Vol. xli. (1897), No. !©• 5 



obtain in this way an infinite assemblage of points P. But 

 we are not entitled to assume that this assemblage, even 

 in the case of a continuous function, constitutes a curve, in 

 the ordinary acceptation of the word. Indeed, one of 

 the most remarkable achievements of the modern Theory 

 of Functions is the discovery of continuous functions 

 possessing properties which transcend our faculties of 

 even mental representation. 



5. After these preliminaries, which have necessarily 

 consisted in a recapitulation of known matters, we can 

 proceed to the theorems more especially in view. 



I. A continuous function cannot change sign without 

 passing through the value zero. 



Let <f>(x) be a function of x which is continuous from 

 x = a to x=b, where b>a, inclusively; and let us suppose, 

 for definiteness, that <p(a) is positive, and <p(b) negative. 

 In the geometrical representation, let OA=a, OB = b, 

 AH = <p(a), BK = <p(b). In virtue of the continuity of 



Y 



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4 



O 



t 



* c 



1 



B X 



</>(x) there is a certain range extending to the right of A , 

 at every point of which <j>(x) differs from AH by less than 

 AH, and is therefore positive. Similarly, there is a certain 

 range extending to the left of B at every point of which 

 <p(x) is negative. Hence the points of the range AB, 



