Outline of a Theory of Functions of an Abstract Variable 77 
state some of previous assertions in a new manner, as for in- 
stance, the remarks following T.78 tell us that a power series 
has at least one singularity on its sphere of convergence. 
For any point in the space Ri an extended regular function 
is regular, singular, or is such that it is not defined at all in the 
neighborhood of the point. The fact that this last condition may 
hold means that we have not been able to continue the given 
function everywhere, and therefore that there may be some 
boundary beyond which we cannot pass. 
Such a boundary is called a natural boundary of the function. 
By confining the range of the argument of a given power se- 
ries to some complex-plane x =a, with a variable complex pa- 
rameter £, it may be found, perhaps, that as a function of € in 
this plane, the power series has a natural boundary. If, more- 
over the power series is a function on to the complex number 
system, we have an ordinary analytic function f(£), whose do- 
main of existence is a region in the a complex-plane, at least 
partly bounded. With the process of continuation as just de- 
veloped for the abstract variable it may be possible to extend 
the definition of f(£) beyond its natural boundary in the complex 
domain by means of a route that passes outside of the a com- 
plex-plane. This process would afford the same kind of gen- 
eralization as that involved in passing beyond an interval of 
convergence in the domain of the real variable and into the do- 
main of the complex variable. : 
However, since there are any number of abstract analytic 
functions which have identical values throughout a given com- 
plex plane, this way of extending a function of a complex 
variable beyond its natural boundary is not unique, unless one 
chooses for some good reason, a particular abstract power series 
as a generalizing function. Perhaps in choosing a power series 
which are solutions of a generalized form of Laplace's differen- 
tial equation, one may find good reason in the analogy with the 
complex power series. 
We can be certain that the radius of convergence pz, Con- 
sidered as a function of #, is not continuous. (If it were, con- 
tinuation beyond a natural boundary in a complex-plane would 
be a fortiori impossible). Consider the following example. 
Let E.) be the space of functions continuous in the interval 
(0, 1), E,o) the space of complex numbers, and let the norm of 
x=x(t) be defined as maxos;s:|x(t)|. In particular consider 
the one parameter set of functions, or unit elements, ¢*, where 
