18 VARIABLES AND FUNCTIONS. [iNT. II. 



If a limit exists on one side but not on the other, the function is 

 said to have a derivative on one side. If no limit exists at the 

 point x, the function has no derivative at that point; e.g. the 



function sin- has no derivative at the point x = Q, although the 

 x 



derivative at any other point however near is 







1 1 



- cos - , 



a? x 



which does not approach a limit as x approaches 0. The last 

 function defined in 8 has no derivative anywhere. We may 

 also find transcendental functions defined by analytic expressions, 

 which nowhere possess a derivative. The function proposed by 

 Weierstrass, 



n = 



/(#) = 2 b n cos (a n 7rx), where < b < 1 ; a is an odd integer, 

 =o 



may be shown to have nowhere a derivative*. 



12. Functions of two or more Variables. If two real 

 variables x and y vary continuously in the respective intervals 



and if to every possible pair of values of x and y is assigned a 

 value of a quantity u, u is said to be a function of x and y. For 

 any particular value of x, u is a function of y, and for any particular 

 value of y, u is a function of x. Suppose that for a certain value 

 y, u considered as a function of x approaches a limit as x approaches 

 a. This limit will in general depend upon the value of y, let us 



call it 



lim u <E> (y). 



x=a 



It may again approach a limit as y approaches a value b. If we 

 consider the limit approached by u considered as a function of y, 

 we shall have in general a function of x, 



lim w = (;). 



y=b 



If x then approaches a, we may have a limit, which is not neces- 

 sarily the same as before, 



lim j lim u ) = lim <!>(;?/) =.4, lim j lim u I = lim "9 (x) B ; 



y=b I x=a 1 y=b x=a \ y-~b > x=a 



* Weierstrass, Abhandlungen aus der Functionenlehre, p. 97; Harkness and 

 Morley, Theory of Functions, p. 58. 



