﻿CONTINUOL S MTNCTIONS 



63 



we can limit this region still more by a line 

 iej) whose equation is ii=2.8. 



Discussion. It is well to notice some pecu- 

 har facts about this function and the proof. 



1. All the conditions placed upon a and 6, 

 excepting that 



0 <h < U 



are sufficient conditions, nothing having been 

 said about their being necessary conditions. 

 This is important because, since it is true, it 

 may be possible to show that some of the re- 

 strictions are unnecessary. 



2. This type of function differs from ordi- 

 nary functions not in having points at which 

 no derivative exists; such functions are com- 

 mon, and such points are called cusps; but in 

 this function every point has this property, that 

 is, we may think of the curve as made up of 

 nothing but cusps. 



3. 



x' is alwavs equal to -^^ 

 a'" 



This is a restriction on the nature of the incre- 

 ments of the variable used in determining the 

 jderivatives. Wiener shows that by placing 

 different restrictions on /\x we get different 

 results. His discussion of Weierstrass's Func- 

 tion is important in that it shows that our con- 

 clusions regarding the derivative depend en- 

 tirely upon how we choose the increment on x. 

 This is a very peculiar property of the function. 



Wiener's results for the cosine function can 

 be developed for the sine function, with the 

 same change in the restrictions as was needed to give Weierstrass's 

 proof of the sine function. 



