8 Lamb, on Continuity. 



Similarly, there will be a certain range, extending to the 

 right of M 1} but not reaching to B, at every point of 

 which y is not greater than /m-^or, let M 2 mark the 

 extremity of this range. Proceeding in this way we get 

 an ascending sequence of points 



M l9 M 2 , M n , . . . 



the property of M n being that at al) points to the left of 

 it y is not greater than jul-^o; whilst every point to 

 the right of M n will have points to the left of it at which 

 this condition is violated. At M n itself we itself we 

 must have 



y=fi-in(r. 



By the reasoning of § 2, the sequence 



M l9 M 2 , M 3 , M^ ... 



will have an upper limit M (say). Moreover, at this 

 point M we must have 



exactly. For, if not, let y 1 be the value of y at this point, 

 and let y' be a quantity between y 1 and /ul. Then in virtue 

 of the continuity, there will be a certain range extending 

 to the left of M for every point of which y<y'. But by 

 the preceding argument, any such range will contain an 

 infinite number of points belonging to the above 

 sequence, and will, therefore, contain points for which 

 the value of y differs from jul by as little as we please, and 

 for which therefore y>y'. The contradiction shows that 

 ji cannot differ from /ul.* 



6. The above investigations have been clothed in a 

 geometrical form, and it remains to consider how far this 

 affects the essence of the demonstrations. 



* The diagram is intended merely to exhibit the mode in which the 

 successive points M lt M 2 , M 3 , . . . are determined. If OK=/x, then, in the 

 figure, Ni bisects HK 2 , N 2 bisects Ni K, and so on. 



For a function which can be adequately represented by a curve, the 

 proof is superfluous, as already indicated. 



