s 



Lamb, on Continuity. 



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

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

 which y is not greater than jul — ^ct, let M 2 mark the 

 extremity of this range. Proceeding in this way we get 

 an ascending sequence of points 



Afi, M 2 , .... M n , . . . 



the property of M n being that at all points to the left of 

 it y is not greater than jm — J n <r, 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—/j.— \n<r. 

 By the reasoning of § 2, the sequence 



Mi, M 2 , M 5 , M n , . . . 



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

 point M we must have 



y=fj-> 



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

 and let y' be a quantity between y l and ju. 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 jm by as little as we please, and 

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

 y x cannot differ from /x.* 



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 x , M 2 , M 3 , . . . are determined. If 0K — ^ t then, in the 

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



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

 75roof is superfluous, as already indicated. 



