42 DEFINITE INTEGRALS. [INT. III. 



Hence F (x) is a continuous function of x. If a is any number 



fx r x 



between a and 6, f(x)dx differs from f(x)dx only by a con- 



Ja J a 



stant, namely the value of the integral I f(x) dx = C. The 



J a 



function F (x) + C is called the indefinite integral of /(#). 



Suppose that h approaches zero either positively or negatively, 

 and let f(x) either be continuous at x, or have an ordinary discon- 

 tinuity, i.e., by making a finite jump. 



Then for any positive number e however small we can find a 

 number h^ of the same sign as h, such that for every x in the 

 interval x, x + h^ (at most excepting x), the value of f(x) for any 

 point differs by less than e from f(x + 0) or f(x - 0), according as 

 h is positive or negative. 



Therefore the value /() in the expression 



differs from f(x 0) by less than e and we have 

 lim J > < + *>-*<'>-/(. + 0) H^W)-^) 



rx 



That is, the integral I f(x) dx F(x) is not only a finite and 



J a 



continuous function of x in the interval ab, but it has at all points 

 where f(x) is continuous a finite and determined derivative f(x) 

 and where f(x) has an ordinary discontinuity, though not having 

 a determined derivative, F (x) has one on the right and left 

 respectively equal to f(x + 0) and f(x 0). If however f(x) has 

 a discontinuity of the second kind, at x, the value of 



h 



as h decreases does not approach a limit and F (x) has no deriva- 

 tive at x. 



The principle here proved enables us to calculate the definite 

 integral whenever we can find a function F(x) whose derivative is 

 /(#), for then 



i. 



The definition usually given of the definite integral, as deduced 

 from the indefinite integral by the above formula, is unsatisfactory, 



