Page 39 - Vector Analysis

§2.2 Limits and Continuity 35

Example 2.10. By Proposition 2.9,

lim x ´ xy + 3 = 0 ´ (0)(1) + 3 = ´3 .
x2y + 5xy ´ y3 (0)2(1) + 5(0)(1) ´ (1)3
(x,y)Ñ(0,1)

Example 2.11. Let f : (0, 8) ˆ (0, 8) Ñ R be given by f (x, y) = x2 ´ xy . We can-
? ´ ?y

x

not apply Proposition 2.9 to compute the limit lim f (x, y), if the limit exists, since

lim (?x ´ ?y) = 0. Nevertheless, if (x, y) (x,y)Ñ(0,0)

(x,y)Ñ(0,0) ‰ (0, 0),

f (x, y) = x2 ´ xy = x(x ´ ? + ?y) = ? ?y) ;
?x ´ ?y y)( x x( x +
(?x ´ ?y)(?x + ?y)

thus Proposition 2.8 and 2.9 imply that

lim f (x, y) = lim ? + ?y) = 0 .
x( x

(x,y)Ñ(0,0) (x,y)Ñ(0,0)

Definition 2.12. Let A Ď Rn be a set, and f : A Ñ Rm be a vector-valued function. The
function f is said to be continuous at x0 P A X A1 if lim f (x) = f (x0). In other words, f

xÑx0

is continuous at x0 if

@ ε ą 0, D δ = δ(x0, ε) ą 0 Q }f (x) ´ f (x0)}Rm ă ε whenever }x ´ x0}Rn ă δ and x P A .

If f is continuous at each point of B Ď A X A1, then f is said to be continuous on B.

Remark 2.13. 1. The notation δ = δ(x0, ε) means that the number δ could depend on x0
and ε.

2. Another way of interpreting the continuity of f at x0 is as follows: f : A Ñ Rm is
continuous at x0 P U if
()
@ ε ą 0, D δ = δ(x0, ε) ą 0 Q f B(x0, δ) X A Ď B(f (x0), ε) .

3. If A = U is an open set, we can assume that δ is chosen small enough so that
B(x0, δ) Ď U in both Definition 2.7 and 2.12. In other words, lim f (x) = b if

xÑx0

@ ε ą 0, D δ = δ(x0, ε) ą 0 Q }f (x) ´ b}Rm ă ε whenever 0 ă }x ´ x0}Rn ă δ ,

and f : U Ñ Rm is continuous at x0 P U if

@ ε ą 0, D δ = δ(x0, ε) ą 0 Q }f (x) ´ f (x0)}Rm ă ε whenever }x ´ x0}Rn ă δ .