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.

34

Differential quotient of a function.

Bk. I. ch. V.

the increase A y in the function. The quotient of differences has the
following properties: when f is a continuous function of x, the quotient
of differences is for any finite value of A x a continuous function of x;
but on the same hypothesis it is also, secondly, a continuous function
of A x, as long as we restrict ourselves to finite but arbitrarily small
values of Ax (see last Section).
When we now endeavour to determine a measure not for an
arbitrary interval but for one point of the function, we have to make
the interval Ax converge to zero. The quotient on the right then
becomes infinitely great, unless its numerator f(x +- Ax) - f(x) also
turn out to be a series of numbers having zero for a limit (~ 10).
But assuming that this is the case, let us consider a point in whose
immediate neighbourhood the unique function is continuous. What
conditions must be fulfilled in order that for continually diminishing
values of A x, the quotient
f(x + Ax) - f(x)
Ax
may present a continuous sequence of numbers tending to a determinate limiting value: zero, finite or infinitely great? We can
again give expression to the fact that we are passing through the
interval Ax in the positive or negative sense, by considering Ax as
a fixed arbitrarily small but finite value, and introducing a number 0
which moves continuously between the limits - 1 and + 1; then our
present object is, to find whether
f(x + OAx) - f(x)
OAx
has a limiting value when 0 converges from - I to zero, or from
+ 1 to zero. The limiting values, arising in the two cases, can be
different; in the first we call the quotient of differences regressive,
in the second, progressive. Upon the latter we fix our attention:
The particular value zero will occur as limiting value, provided,
for each number d however small, a number Ax can be found such
that the absolute amount of this quotient for every value of 0 is
less than 6. Here the numerator can change sign arbitrarily, i. e.
f(x - OAx) can be at one time greater, at another less than f(x),
or in other words: Provided zero is the limiting value of the quotient
of differences, the function f(x) can, in the neighbourhood of the point x,
oscillate arbitrarily many times about the axis of x.*)
*) The fluctuations (differences of ordinates), we remark only in passing, are
infinitely small of an order higher than the first (Chap. VII). The finction x2 sinx
has ex. gr. for x = 0 the differential quotient 0, for we have
Lim Jf(o +- h) - f(o) = Lim ih sin = 0
}Ii l si 1SJ-0