hit 
FP d(#)| <r, if Mory d, 
and this last result in which the initial function u, (w) occurs no 
more, says: Corresponding to any arbitrarily little amount t there is 
an amount gd, such that the absolute value of the transmuted of a 
function, in the arbitrary point z of the N.F.O. is smaller than r 
if the absolute value of that function itself is any where in the N.F.F. 
smaller than d, provided that function belongs to the F.F. consi- 
dered. Hence: if the distance between two functions u, and u, of 
the F.F. is smaller than d, we have in the whole N.F.O. 
Tu, (a), (a))| <r 
or, according to the additive property 
| Tu, («)—T u, (©)| <r. 
The proposition has thus been established. 
The expressions “continuous in a point of the F.F.” ; “continuous 
in the F.F.”; „uniformly continuous in the F.F.” can consequently 
be substituted for each other; as a rule we shall make use of the 
middle one. Or, if we want to direct our attention at the same 
time ot the numerical field, we shall use an expression like the 
following: The transmutation is continuous in the pair of fields 
considered. 
11. The mode of reasoning followed in the preceding proof suggests 
the observation that the discussions about the.continnity of an addi- 
tive operation may be simplified by using the following proposition : 
If an additive operation is to be continuous, it is necessary and 
sufficient that there is corresponding to any arbitrarily chosen number 
t a number d, such that in the whole N.F.O. 
Tul) 
under the single condition that 
MZ, 
Here M, denotes again the maximum modulus of « in the 
N.F.F. 
The condition contained in the proposition is sufficient. For, 
if it is fulfilled, and if the distance between two functions u, and 
u, of the F.F. is at most equal to d, the transmuted of their difference 
will, in absolute value, in the whole N.F.O. be at most equal to r and 
the same will therefore, according to the additive property of 7, 
be the case for the absolute difference of the two transmuted. 
The condition is necessary. If it is not fulfilled, this means 
that there exists a positive number r such that there is corresponding 
