Mathematics. — "On the Points of Continuity of Functions" . Bj 

 Prof. J. Woi.FF. (Coinmiiiiicated by Prof. Hendrik dk Vkiks). 



(Communicated at the meeting of February 24, 1923). 



Let f[P) be a function of llie coordinates of a point F in a 

 space with an arbitrary number of dimensions. The points wliere 

 ƒ is continuous, form an inner limitinq set, i.e. the intersection of 

 an enumerable set of open sets of points i2„, where we may assume 

 that i2„-|-i is a part of i2„ for any n. For the points, wliere tlie 



function oscillates less than -, form an open set i2„ because tiie oscil- 



n 



lation is an upper semi-continuous function. The set of the points of 



continuity is the intersection of all i2„, ?t ^= 1, 2, 3, . . . . Young ') has 



shown that to any inner limiting set £■ given in a linear interval, there 



belongs a function in that interval which is continuous in the points 



of E and discontinuous in any other point. We siiall give here a 



simple proof, which is directly valid for spaces of any number of 



dimensions. 



1. Let a set of points E be given as the intersection of an enu- 

 merable set of open sets 52,,, where £in + i is a part of (or coincides 

 with) i2„. 



We define f{P] for any point of space in the following way : 

 in the first place f(P) = if P lies in E. Now let P be a point 

 not lying in E, Up the least value of n for which ii„ does not 

 contain the point P. 



We put 



where »p(P) is the function which in the points of space of which 

 all the coordinates are rational, is equal to 1, in any other point 

 of space equal to — 1. 



We may say that (1) holds also good for the points of E, if 

 there we assume np -^ cc . 



2. Now we shall show, that f{P) is continuous in the points of 

 E and discontinuous outside them. 



1) W. H. Young. Wiener Sitzungsber., vol 112, Abt. II>, p. 1307. 



13 

 Proceedings Royal Acad. Amsterdam. Vol. XXVL 



