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.

~ 100. Condition of the Theorem of the Mean Value for a continuous function. 171
extremities x0 and X of the interval, and its differential quotient is
continuous as that of f is; therefore there is a point
X0 + O (X- x),
at which:
1 { xo + 0(X- xo)} fi{x + (X — xo)} -K J.=
The equation:
/'(X) - f(c) f/x o- oh) f(x)
X —xo ft x0 o+ -(X-x,), or: (x + O),
holds for every value of h, when x and x + h denote any points in
the interval. For an arbitrarily small prescribed value of h, we can
choose x so that x + h may represent any point x, in the interval.
Hence we have the result: For values of h however small, the equation
f (x) - f(xl - h)
f (x -O Lh)
can be fulfilled at every point x, in the interval. Making h converge
to zero, while maintaining the value x,, the right side passes over
continuously into f (xl), therefore the regressive differential quotient is
identical with f1 at every point, as was to be proved.
Accordingly the Theorem of the OMean Value holds for a continuous
function if its progressive differential quotient is likewise continuous;
and hence follows: A continuous function, whose progressive differential
quotient vanishes throughout any interval, is constant in this interval.
We shall further for completeness deduce the uniform continuity
of the quotient of differences. It has to be proved, that in consequence of the continuity of f and f1 for every value of x, a superior
limit can be assigned for h and A x, such that for all smaller values
F f(x + +Ax) - f(x+h) f(x +Ax)-f(x)]
L Ax Ax
remains smaller than an arbitrarily small number J.
The first quotient can be brought to the form f/ (x + h + OA x),
the second is equal to fl(x + 0'Ax). Since f1 is continuous, we are
able merely by choice of h and Ax to make the difference
f (x + h + q {Ax) - f (x + -O'Ax)
smaller than 6.
Thence it follows, that if the function f(x) and its derivate f; (x)
be defined for an entire interval from a to b, a superior limit can -be
assigned for Ax, sufficient, for a given value of 6, that every smaller
interval Ax between a and b shall satisfy the inequality
f( + Ax) f(x) f () <
Ax f
For if, while x converges to a value x', Ax were to fall below any
assignable limit, arbitrarily near this point it would become impossible
to satisfy the inequality: