324 Mr. P. E. B. Jourdain on the Transfinite 



Suppose, firstly, that we have to deal with real one-valued 

 functions of one real variable, and both the argument and 

 function values are certain continua ; say 



where a. h, c, d are any real numbers, which may be + x» . 

 Then e = b = l, a='D = '2^o = r, and (1) becomes 



I£, however (as iu the case with continuous, analytic, . . . 

 functions) , the D = ?*^ — since the function is completelv 

 determined for every argument x, by its datum for a certain 

 enumerable aggregate among these argument values — then (1) 

 becomes 



Thus the aggregate of all real continuous"^ or analytic 

 functions of one real (or complex) variable is at most of the 

 cardinal number of the continuum. 



That, further, these aggregates are also at least of cardinal 

 C follows from the fact that there is a partial aggregate of 

 this cardinal [e. o., linear functions) . Hence by the Schroder- 

 Bernstein theorem, the cardinal is exactly C. 



These two results have been known, but their derivation 

 from the general formula (1) appears interesting. 



2. 



The property that their cardinal number is C is pos- 

 sessed by all functions to which an '' existence-theorem " is 

 applicable. For such functions, when they cannot be con- 

 structed by a finite number of operations on the variables 

 and undetermined constants (L e. when they are not rational 

 functions), must be representable as the limiting function of 

 a fundamental sequence of functions which are either rational 

 or limits of a sequence of rational functions. Kow the 

 cardinal number of the aggregate of all functions defined by 

 fundamental series (an enumerable aggregate) is 



This result includes all Cantoris results, except the one 

 treated above — the case C^, — and that on the aggregate of 

 integrahle functions. This last we shall now consider. 



* Or even of functious continuous except for an aggregate of cardinal i^. 



