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tions that have the same circle of convergence, but about such as 
belong to the same circle. Moreover we need not use now the arti- 
ficial definition of Frécuer for the distance of two functions; we 
shall simply understand by it the maaz/muwm modulus of their differ- 
ence in the N.F.F. In the same way we mean by the distance of 
the results which the operation produces for two functions the 
maximum modulus of the difference of those results in the N.F.O. 
If we understand by a point of the F.F., a function wu, (#) with 
which the independent variable function u (ez) may be identified 
we can say that the notion of continuity only appears in limiting 
points, ie. in “points” w, («), which have the property that at 
an arbitrarily small distance of them — or in an arbitrarily 
small vicinity as we might say still “points” wu («) of the F.F. 
are found. As a rule all “points” of the F.F. have that property, 
in other words, there are no isolated “points” in the F.F. or: the 
F.F. is dense in itself. 
According to the classic definition of continuity an operation will 
be continuous, if, generally speaking, the distance between the results 
which it produces for two functions of the F.F., becomes smal/ 
with the distance of these functions themselves. More amply and 
exactly our definitions of continuity read as follows: 
1. A transmutation 7’ determined in a pair of associated fields 
is called continuous in a “point” w, («e) of the F.F. if there is cor- 
responding to each arbitrarily given number rt, however small, a 
number J, such that for all the values of 2 in the N.F.O. 
Tule) Tue) |<t 
under the single condition that 
| u (2) —u, (a) | <I 
for all the values of « in the N.F.F. 
2. A transmutation 7, defined as before, is called continwous 
in the F.F., if, according to the preceding definition, it is continuous 
in any point of the FF. 
3. A transmutation 7’, defined as before, is called wniformly con- 
tinuous in the F.F., if corresponding to any arbitrarily given amount 
t, however small, there is an amount d such that for any “pair of 
points” u, (w) and wu, (#) of the F.F. and for all z-values of the N.F.O. 
| Tu, («) — Tu, (a) | rt 
under the single condition that 
at, (a) — u, (2) | <d 
for all the values of zin the N.F.F. 
If in statements about the continuity of a transmutation we 
