1110 
want to draw at the same time our attention to its numerical field, 
we shall say for the sake of completeness: “continuous for, or with 
regard to or even in the N.F.O.” In using the preposition “in” it 
should be borne in mind that there is no question about a comparison 
of the results of a transmutation, for one and the same function, in 
a point of the N.F.O. and a numerical neighbourhood of that point; 
the notion of continuity of a transmutation merely refers to a 
comparison between the results for one and the same point of the 
N.F.O. in a “point” of the F.F. and a functional neighbourhood of 
that “point”. 
The form given here for the definition of the continuity of a 
transmutation corresponds to that of Caucny for the continuity of 
functions. Equivalent to it is the following form corresponding to 
that of Heine for the continuity of functions: 
A transmutation is continuous if the result v(#) = Tu («) in the 
N.F.O. approaches to v, (*) = Tu, (x), if the function w, («) in the 
N.F.F. approaches to u, (#). lt is meant by it that if u («) as the variable 
function is identified successively with the functions of a convergent 
sequence 
OLED IAN Cc) GP rt irl EA Ngee 
which all belong to the F.F. and in the N.F.F. uniformly approach 
to uw, (), the transmuted v (#) = Tu (x) is identified successively with 
the functions of a sequence . 
Sea Gc) EN esl (al ec 
which converges in the N.F.O. uniformly towards »v, (#), and that 
this happens for any suchlike fundamental series of functions uw, (2). 
10. It is of importance to observe now that the difference 
between the three cases of continuity mentioned in the preceding 
section is superfluous for additive operations, since the following 
proposition’ holds, which shows analogy with the well-known theorem 
of Heine from the theory of functions: 
An additive operation, which is continuous in only one point u, (x) 
of the functional field, is uniformly continuous in that field. 
Let J/, be the upper limit of the modulus of the function w in 
the N.F.F. It is to be derived then from the hypothesis that, 
corresponding to an arbitrarily given amount t, there is an amount 
J, such, that in an arbitrary point « of the N.F.O. 
Tu (x) — Tu, (#) | <x tif only M, —w eel J. 
Suppose 
aa) =a) i) 
then we have, from the additive property of 7’ 
