CHAPTER II. 



VARIABLES AND FUNCTIONS. 



8. Functions. A real quantity is said to vary continuously 

 between two values a and b if it assumes successively all real 

 values, rational and irrational, comprised in the interval between 

 and including the values a and 6. The notion of continuity was 

 arrived at by considering the motion of a point which at successive 

 instants of time occupies the positions of all the points between 

 those representing a and b, and by the nature of motion cannot 

 omit any intermediate value. 



A quantity y is said to be a function of a variable x, in an 

 interval from a to b, if for every value that x may take in the 

 interval ab, there is assigned a definite value of y. A function 

 defined in this somewhat restricted manner is called uniform, or 

 one-valued. We may extend the definition so that for each value 

 of x t y may have several values, in which case it is said to be a 

 multiform, or many-valued function of x. This definition, due to 

 Dirichlet, is independent of the question whether we can find an 

 analytic expression for the value of y in terms of x or not. For 

 example the analytic expressions 



( i ) a + OjX 4- a< 



( \ 



(3) 7# - a, 



(4) e*=l + 



