14 VARIABLES -AND FUNCTIONS. [INT. II. 



a? a? 



(7) 



x x 21 x* 3! x s 



are all functions of # in any interval from a to b, where a and 6 are 

 both positive or negative finite numbers different from zero. Of 

 these the first three are examples of the class called algebraic 

 functions, or such as are defined by an algebraic equation between 

 x and y. An algebraic equation is one in which only a finite 

 number of powers with finite integral exponents and products of 

 such powers of the variables appear. All other functions are called 

 transcendental, and the last four above are examples of such. All 

 the above are uniform functions, except (3), which has two values, 

 one of which is the negative of the other. A function such as (i) 

 is called a polynomial, or a rational entire or integral function. A 

 function such as (2), or the quotient of any two polynomials, is 

 called a rational fractional function. (3) is an example of algebraic 

 irrational functions. (4) and (6), being defined by convergent 

 infinite series of positive powers of x, are called integral trans- 

 cendental functions, and the quotient of two such is called a 

 fractional transcendental function. The distinction between 

 rational and transcendental functions is similar to that between 

 rational and irrational real numbers, depending on the matter 

 of finiteness or infinity in the method of specification. 



A uniform, continuous, integral, rational or transcendental 

 function is called holomorphic. 



A function taking the value 1 from the value x 0, inclusive, 

 to x = 1/2, exclusive, the value 2 from x 1/2, inclusive, to x 3/4, 

 exclusive, the value 3 from x 3/4, inclusive, to x = 7/8, exclusive, 

 tc., and a function defined as taking the value 1 for all rational 

 points, and 2 for all irrational points, would be, the first difficult, 

 the second probably impossible to define by analytic expressions. 

 The former, being perfectly defined for every real value of x from 

 .zero to 1, excluding the latter, satisfies the definition of a function 

 in that interval, while the latter satisfies it in any interval. 



9. Limit of a Function. If y =/(#) is a function of the 

 continuous variable a? in a certain interval including the value x=a, 

 and if there exists a number A having the property that to any 



