Page 128 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 6] PARTIAL DERIVATIVES 119

ITERATED LIMITS

The iterated limits lim lim f ðx; yÞ and lim lim f ðx; yÞ , [also denoted by lim lim f ðx; yÞ and
x!x 0 y!y 0 y!y 0 x!x 0 x!x 0 y!y 0
lim lim f ðx; yÞ respectively] are not necessarily equal. Although they must be equal if lim f ðx; yÞ is to
y!y 0 x!x 0 x!x 0
y!y 0
exist, their equality does not guarantee the existence of this last limit.
x y x y x y
EXAMPLE. If f ðx; yÞ¼ ,then lim lim ¼ lim ð1Þ¼ 1 and lim lim ¼ lim ð 1Þ¼ 1. Thus
x þ y x!0 y!0 x þ y x!0 y!0 x!0 x þ y y!0
the iterated limits are not equal and so lim f ðx; yÞ cannot exist.
x!0
y!0

CONTINUITY
Let f ðx; yÞ be defined in a neighborhood of ðx 0 ; y 0 Þ [i.e.; f ðx; yÞ must be defined at ðx 0 ; y 0 Þ as well as
near it]. We say that f ðx; yÞ is continuous at ðx 0 ; y 0 Þ if for any positive number we can find some
positive number [depending on and ðx 0 ; y 0 Þ in general] such that j f ðx; yÞ f ðx 0 ; y 0 Þj < whenever
2
2
2
jx x 0 j < and jy y 0 j < ,or alternatively ðx x 0 Þ þð y y 0 Þ < .
Note that three conditions must be satisfied in order that f ðx; yÞ be continuous at ðx 0 ; y 0 Þ.
1. lim f ðx; yÞ¼ l, i.e., the limit exists as ðx; yÞ!ðx 0 ; y 0 Þ
ðx;yÞ!ðx 0 ;y 0 Þ
2. f ðx 0 ; y 0 Þ must exist, i.e., f ðx; yÞ is defined at ðx 0 ; y 0 Þ
3. l ¼ f ðx 0 ; y 0 Þ
If desired we can write this in the suggestive form lim f ðx; yÞ¼ f ð lim x; lim yÞ.
x!x 0
x!x 0 y!y 0
y!y 0

EXAMPLE. 3xy ðx; yÞ 6¼ð1; 2Þ ,then lim f ðx; yÞ¼ 6 6¼ f ð1; 2Þ. Hence, f ðx; yÞ is not contin-
0
If f ðx; yÞ¼
ðx; yÞ¼ ð1; 2Þ
ðx;yÞ!ð1;2Þ
uous at ð1; 2Þ.Ifwe redefine the function so that f ðx; yÞ¼ 6for ðx; yÞ¼ ð1; 2Þ,then the function is continuous at
ð1; 2Þ.
If a function is not continuous at a point ðx 0 ; y 0 Þ,itis said to be discontinuous at ðx 0 ; y 0 Þ which is then
called a point of discontinuity. If, as in the above example, it is possible to redefine the value of a
function at a point of discontinuity so that the new function is continuous, we say that the point is a
removable discontinuity of the old function. A function is said to be continuous in a region r of the xy
plane if it is continuous at every point of r.
Many of the theorems on continuity for functions of a single variable can, with suitable modifica-
tion, be extended to functions of two more variables.

UNIFORM CONTINUITY
In the definition of continuity of f ðx; yÞ at ðx 0 ; y 0 Þ, depends on and also ðx 0 ; y 0 Þ in general. If in a
region r we can find a which depends only on but not on any particular point ðx 0 ; y 0 Þ in r [i.e., the
same will work for all points in r], then f ðx; yÞ is said to be uniformly continuous in r.Asin the case
of functions of one variable, it can be proved that a function which is continuous in a closed and
bounded region is uniformly continuous in the region.

PARTIAL DERIVATIVES
The ordinary derivative of a function of several variables with respect to one of the independent
variables, keeping all other independent variables constant, is called the partial derivative of the function
with respect to the variable. Partial derivatives of f ðx; yÞ with respect to x and y are denoted by