An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 127 —128.

Deduction of its continuity.

221

hence for any d however small, a value h can be found for which:
abs [F(x + h) - F(x)] < 6.
Accordingly the criterion of continuity is satisfied.
The theorem we have proved can also be stated thus: If the
function expressed by the series be discontinuous at a point of its convergency, the series must converge unequably in the neighbourhood
of this point.
From the theorem it follows: If the series converge uniformly
without exception in its convergency, it expresses a function everywhere
continuous in the same interval.
These theorems admit of conversion only on a certain hypothesis.
When the infinite series expresses a continuous function at a point,
to any point x belongs a finite value n such that the remainder
R,. and all that follow it are less than 6; moreover a value of h can
be assigned for which:
abs [F(x + h) - F(x)] < ---
and likewise:
abs [Z(x + ) - I(x)] < -~
Accordingly from the equation:
F(x + h) - F(x) = (x + h) - L(x) + R, (x - Vh) - n (x)
it follows that:
abs [RA (x + h)] also < 4.
But it does not follow from this, that all the following remainders
in the interval + h continue less than 6. This will only be certainly
the case when all terms have the same sign in the interval x to x + h,
because then the amounts of the remainders form a decreasing series.
Accordingly the statement of the theorem is:
If in the neighbourhood of a point a finite value n can be assigned
such that the nth and all following terms in the series retain the
same sign in the interval, then the uniform convergence of the series
is a consequence of its continuity at this point; or again: If an
infinite series converge absolutely in the neighbourhood of a point,
its uniform convergence is a consequence of its continuity at this
point. *)
*) It can be shown as a fact that the continuity of a convergent series
alone is not a sufficient condition for its uniform convergence, by examples of
continuous but unequably convergent series recently formed by Du Bois-Reymond,
Darboux and Cantor. An example given by Cantor is: The infinite series having
its general term:
' () = x x -_ (n -- 1)
fn(x n2 -- 1 ( )2- +X~ 1