﻿CONTINTJOTTS FUNCTIONS 



59 



7. A Continuous Function Having Nowhere a Derivative. 



By Rainard B. Bobbins, A. M. 



It had been generally assumed until the latter part of the nine- 

 teenth century that a continuous function of a single variable has 

 a definite, finite derivative for all values of the variable except 

 for isolated singular points. There is nothing in the writings of 

 Gauss, Cauchy, or Dirichlet to indicate that they doubted this as- 

 sumption\ It seems that Riemann was the first to state positively 

 that this asumption was wrong. He made this statement about 

 1861, or possibly earlier, to some of his students and gave as a 

 function for which the assumption was wrong, 



1 n 



His proof has been lost. Several mathematicians have since dealt 

 with continuous functions having no derivative at any one of an 

 infinite number of points in an arbitrary small interval of the vari- 

 able, but AVeierstrass^ was the first to find a function having a de- 

 rivative for no value of the variable. The function which he con- 

 sidered is represented by the series 



f(x)=^n COS (a^'iTTt) 



in which a is an odd integer, and o<Ch<il. When the product ah 

 exceeds a certain limit, f(x) has no derivative. AA^iener^ has since 

 written the most complete discussion of this function that has as yet 

 appeared, presenting it analytically and geometrically. 



The purpose of this paper is to show that another function, 



f(x)=in h^'sin (a'^xn) 



0 



has the same properties with regard to the derivative as the func- 

 tion discussed by AVeierstrass and AViener. when restrictions are 

 placed on a, and 5, slightly different from those which they used. 

 The same method will be used as was used by AVeierstrass. The no- 

 tation wdll not be essentially different from that of AVeierstrass. 

 but in a few cases will be somewhat simpler. 



To prove that the function f: n h*' sina^^xn in ivMch o < b < 1 



0 



and a is odd, lias no derivative when a a/ud h satisfy certain con- 

 < lit ions. 



1 Weicrstra.^s. We. kv, II. 71. 



'.Jon,,ial fiir Mathnnatik. Bd. i>0, 221. 



