136 Mr. W. Williams on the 



although A is infinitely small and F(0-f-#) is continuous 

 between and A, we cannot without a special examination 

 treat F(0 + x) as constant in the integral, and write 



%) — h 2 



For, since -x r must be infinitely small compared with A 2 , 



ii i i. sin(2n + l)i6>, . fl ., 



however small A may be, ja — nas an minute num- 



ber of oscillations between and A. In such a case we must 

 write the integral in the form 



F(4* sin(2 " + 1) ^ + fW+g)-FW] sin( y*V 



J —h 2 %/— A 2 



and determine under what conditions, if any, the second term 

 vanishes. 



24. Now although the function F (6-\-x) is continuous 

 between and A, and therefore ¥(x-\-6) — F(^) is infinitely 

 small between the same limits, it by no means follows that 

 the second term in the above vanishes when n = cc . Its 

 vanishing depends upon the nature of the continuity of the 

 function F, and we have only proved that it vanishes when 

 the continuity is such as to admit of the existence of a derived 

 function F'. In modern analysis, a function F(V) is said to 

 be continuous at the point x if, S and e being positive quan- 

 tities as small as we please, and 0* any positive quantity at 

 pleasure between and 1, we have for all values of 6 

 F(x + 6S) — F(x) less in absolute magnitude than e (Cayley, 

 art. "Function/"' Encyc. JBritt.). In other words, F(#) is 

 continuous at a point x if a region (x — S) to (x-\-8) can be 

 found such that the values of the function for all points within 

 this region (that is, F(x + 68) for all values of between 

 and 1) differ from its value at x by a quantity < e, e being 

 infinitely small : the function may vary in any manner what- 

 soever within this region provided only the difference between 

 its greatest and least values is not greater than e. Hence a 

 function may be continuous according to this definition with- 

 out admitting of a differential coefficient, for the existence of 

 a differential coefficient implies, in addition to the above, that 



lim rF(*+-8)-F(*)n , , , , . , , 



^_ — - — =—£ —*- has everywhere a determinate value, 



or, geometrically speaking, that F(# + 8) — F(x) is ultimately 



* 6 is here the symbol for a positive fraction, and not the variable of 

 integration. 



