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.

88

Functions of more than one independent variable. Bk. 1. ch. IX.

becomes less than the arbitrarily small quantity 2 for all values of
0 and l, merely by the values chosen for h and k.
This way of putting it is important, because it reduces the investigation of continuity of a function with two variables to that of
uniform continuity in regard to each of them.
Examples:
1. The function s =- a xt y1' where u and v are positive integers and
a an arbitrary constant, is a continuous function of -both variables.
For, the absolute amount of
(y + Iq 7), -_- Y VI (+ ~ ) y I -t v2 (+~ k) yv-2.t.. (++ k)) }
is less than
N[7q + (gq) +.. (j) -\ N (17),
where N signifies the greatest among the coefficients within the above
brackets. Assuming qklv < 1, then the amount of the difference
(y/ + -)yV - is < Nr1
Consequently
a (x + )' Ohl [(iy + n 7)"- y" < a (x -t O), " L r _ V N,
and if this is to be less than S for all values of 0 and q, deinoting
by X the greatest absolute value which (x + Oh)7 takes for all
values of 0, we have in order to determine k
t- < —C e. i —
i-ni ^ aXN ' Y <Y ~+aXN
A like consideration shows that the difference
a (y + tY k)" (x -+ 0 h),t - x,] is also < 6 when h < -,
The considerations in this example serve for the proof of the general
theorem:
If f'(x y) = Q)(x) ) (y) where gp and iJ/ are continuous functions of
the variables x and y, then f is a uniformly continuous function of
x as well as of y, i. e. a continuous function of both variables. For
p(x +- Oht) [p (y + '-r1) - P(y)]
can be made < 6, exclusively by choice of k independently of Oh, and
~ (y + - 7) [ p(x + Oh7) - (px)]
can be made < ', exclusively by choice of h independently of,7k.
1
o ~ == - is discontinuous at all points of the right line x = 0)
xy
and of the right line y =- 0. For, is for all values 'of x a discontinuous function of y when y == 0, and for all values of y a discontinuous
function of x when x =- 00