THE KANSAS UNIVERSITY 

 SCIENCE BULLETIN 



VoL.VII, No. 1.] JUNE, 1913. [vrxviirNo^'i. 



THE INTERRELATIONS OF EIGHT FUNDAMENTAL 

 PROPERTIES OF CLASSES OF FUNCTIONS. 



BY 



ARTHUR DUNN PITCHER. 



Preface. 



In his Introduction to a Form of General Analysis* Moore defines 

 General Analysis as "the theory of classes of functions, functional 

 operations, etc., involving at least one general variable." The same 

 memoir contains a theory of classes Tl of real-valued, single-valued 

 functions t ju of a general variable p on a general range '^. This theory 

 relates to certain properties of classes Wl of functions. 



A property of functions n or of classes 3)1 of functions is said to be 

 of general reference in case it is defined for functions or classes of func- 

 tions on a general range '^. Thus jiniteness on the range ^ and con- 

 stancy on the range *$ are properties of general reference of functions n. 

 On the other hand a property of functions ix or of classes 9)J of functions 

 which refers to some special feature of the range "!{5 is a property of 

 special reference. Continuity is a property of special reference of 



* Cf. E. H. Moore, Introduction to a form of General Analysis, The New Haven Mathe- 

 matical Colloquium (Yale University Press, New Haven, 1910), pp. 1-150. In the sequel 

 we refer to this paper as I. G. A. An exposition of the fundamental notions of his theory 

 and its applications is given by the same author in his paper, On a Form of General Analysis, 

 with Application to Linear Differential and Integral Equations, read before the Section on 

 Analysis of the Rome Congress of 1908, Atti, etc., Vol. 2 (1909), pp. 98-114. 



t In this paper the term function is applied only to real-valued single-valued functions. 



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