9 11] VARIABLES AND FUNCTIONS. 17 



greater and greater positive and negative values in any interval, 

 however small, including zero. 



10. Continuity of Functions. A function is said to be 

 continuous at a point x = a, if for any positive e there is a 8 such 

 that 



\f(a + h)-f(a)\<e, \h\<8 



for all values of h whose absolute value is less than S. 



If the condition holds only for positive values of h, the function 

 is said to be continuous on the right, if for negative, on the left of 

 a. A function may be discontinuous at a point by reason of jump- 

 ing abruptly from one finite value to another, becoming infinite, or 

 oscillating through a finite or infinite range in an infinitesimal 

 interval. The last function defined in 8 is nowhere con- 



tinuous, and the next to the last is discontinuous at the points 



i^ 



1/2, 3/4, 7/8, etc., for the first reason, the function e x is discon- 

 tinuous at x = for the second reason, and the functions 



111 



sin - , - sm - , 



XX X 



1 



are discontinuous at the same point for the third reason, e x is 

 continuous at the left, discontinuous at the right of the point #=0. 



A discontinuity arising from a finite jump, or an infinite in- 

 crease or decrease, is called an ordinary discontinuity, while one 

 arising from an oscillation is called a discontinuity of the second 

 kind, and the value of no at which it occurs is called an essentially 

 singular point for the function. 



11. Derivative. In the neighbourhood of any value of the 

 variable x, the difference-quotient 



-f(x) = /(* + h) -f(x) 

 x h 



is a function of the increment h of the variable. If this quotient 

 approaches a limit as h approaches 0, the value of the limit is 

 called the derivative of the function f(x) at the point #, and is 

 denoted by/ 7 (x) or by 



dy = Km /(* + &)"/(*). 

 dx h =o h 



w. E. 2 



