A FEW EXAMPLES IN THE THEORY OF 

 FUNCTIONS 



By Arnold Emch 



i. The progress of the student in a first course of the theory of func- 

 tions is usually retarded by a lack of proper illustrative examples. For 

 a clear understanding of the problems involved in such a study, the 

 beginner must be aided by geometrical representations and models and 

 by exercises fully worked out by the teacher as well as by the student. 



In what follows I shall present a few examples which may be con- 

 sidered according to the foregoing suggestions. 



2. Example I. All rational numbers form an enumerable 



SYSTEM. 



In other words, with every number of the sequence of natural num- 

 bers i, 2, 3, ... we can associate a definite rational number, so that 

 to every rational number corresponds conversely a definite natural 

 number. This relation is also designated as a one-to-one correspondence. 

 As the analytical reasoning for the establishment of this theorem is 

 sufficiently known, 1 I shall not repeat it here and merely give a geometri- 

 cal illustration of it which makes its truth intuitively apparent. 2 



Divide the :ry-plane into a net of equidistant lines parrallel to the x- 



and v-axes ; let the distance between any two parallel lines be unity and 



extend the net indefinitely. At every point of the net write its abscissa 



as a numerator and its ordinate as a demoninator, so that every point 



P 

 of the net is represented by a fraction - which may be positive or negative, 



a proper fraction or an integer. Then, beginning at the origin, describe 

 a spiral-line as indicated in Fig. i and continue it indefinitely. As all 

 rational numbers are represented by the points of the net, it is clear that 

 following the spiral-line all rational numbers will be met in a definite 



1 Borel: Lefons sur la Thiorie des Fonctions. 



• From Prof. Hurwitz's lectures on analytic functions at the Polytechnic of Zurich. 



237 



