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.

~ 52. 53.

Partial derived functions.

91

the partial deriveds lf and fy being supposed to have definite values
ax
in the neighbourhood of the point under consideration?
We have identically:
f(x-+-Ax, y+Ay)-f(x, y) f(x -Ax, y+A y)-f(x, y+Ay)
Ax Ax
+ f(xy+Ay)-f(x, y) Ay
Ay Ax
If in the first quotient on the right we first make Ay vanish,
it becomes -f(x Ax,y) —f(x, y) and this passes over into f(x, J) as Ax
Ax Ox
vanishes; but if reversing the order we first put Ax = 0, we get the
expression f(xy+ A) which will likewise become af/(x, ) for Ay=0,
Ox ax
only when it is a continuous function in regard to y. Now what value
results when A y and A x converge to zero simultaneously in any
manner? In order that af-, ) be again limiting value independently
ax
of the manner of convergence, the condition must be satisfied that a
Ax can be found and independently of it a Ay, such as will make the
absolute amount of the difference
abs ff(x-+Ax,+^ya-f(x,y+Ay) — f(x + ox,y y + Al A y) -f(x, y+rAy)]
asAx OAx
where 6 denotes an arbitrarily small quantity, while the proper fractions 0 and X assume all possible values. This inequality is expressed in the words: The quotient of differences must be a
uniformly continuous function of Ax and of y.
This condition is necessary and sufficient and cannot be replaced by any other. The differential quotient proceeds by continuous
transition from the quotient of differences and we can therefore easily
conclude, that from this requirement the continuity of the function
aLf in regard to y necessarily follows, without this therefore being
fitted to replace the above. For, since the condition must be fulfilled
for all values of Ay independently of the value Ax, it holds also for
Ax=0, i.e.,
af(x, Y+A y) f(xY+, y)< ( <'< 1)
ax ax
If, putting 0 =- 1, V 0, we write the above inequality in the form
asL ---- Ay Ay Ax~,
abs [f(xZAx, Y -1-A Y)-f (x-+]-x, y) f(x,y -]aY) —f(x,y) Y<
since it holds for values of Ax and Ay however small, whose ratio
has an arbitrary finite limiting value k, we see that the continuity
of f/ in regard to x is also involved in the above condition, for
ay