188 



Lel nk first assume that F belongs lo E. In tliis case ƒ (Pj = 0. 

 If' f be an arbitrary positive number, we may choose the natural 

 number r in such a waj that 



'V f (2). 



I' 



As /-* lies in ii, and il, is open, liiere exists a region U round 

 P which lies also in i2,. For any point Q of (I we have therefore 

 iiq ^ J', so tliat accortling to (1) and (2) 



\AQ)\ <^ 



Hence /' is continuous in anv point of E. 



Let us now assume I^ lo lie in tiie complement of £. If F is 

 not an limiting-point of ii„„, it lias a neighbourhood U which has 

 no point in common with i*„ and which lies in 52„ i . For any 

 |»oint (^ of U we have in this case 7iQ = )}p. Hence 



1/(^)1 = \/in\ 



As the points where /' is positive as well as the points where ƒ 

 is negative, lie everywhere dense on U, the oscillation of ƒ in F 

 is equal to 2 \/{Pj \. 



If however P is an limiting-point of i2„ , every neighbourhood U 

 of P contains a part of ii„/,. For any point of that part fif^UQ, 

 hence 



/(«)-yw|>-?^-,^ ... (3) 



As the points Q for which the inequality (3) holds good, have 

 P for a limiting-point, P is a point of discontinuity of /'. Herewith 

 the theorem has been entirely proved. 



I 



