The Real And Complex Number Systems

(d) fx 1/1 e 1/x if x 0, f0 0.
Solution: f is continuous on R 0, and since lim x0
e 1/x and lim x0
e 1/x 0,
we know that f has an irremovabel discontinuity at 0. In addition, f0 0and
f0 1, we know that f has the lefthand jump at 0, f0 f0 1, and f is
continuous from the right at 0.
4.59 Locate the points in R 2 at which each of the functions in Exercise 4.11 is not
continuous.
(a) By Exercise 4.11, we know that fx, y is discontinuous at 0, 0, where
fx, y x2 y 2
if x, y 0, 0, andf0, 0 0.
x 2 y2 Let gx, y x 2 y 2 ,andhx, y x 2 y 2 both defined on R 2 0, 0, we know that g
and h are continuous on R 2 0, 0. Note that h 0onR 2 0, 0. Hence, f g/h is
continuous on R 2 0, 0.
(b) By Exercise 4.11, we know that fx, y is discontinuous at 0, 0, where
xy 2
fx, y
if x, y 0, 0, andf0, 0 0.
xy 2 2
x y
Let gx, y xy 2 ,andhx, y xy 2 x y 2 both defined on R 2 0, 0, we know
that g and h are continuous on R 2 0, 0. Note that h 0onR 2 0, 0. Hence,
f g/h is continuous on R 2 0, 0.
(c) By Exercise 4.11, we know that fx, y is continuous at 0, 0, where
fx, y 1 x sinxy if x 0, and f0, y y,
since lim x,y0,0 fx, y 0 f0, 0. Letgx, y 1/x and hx, y sinxy both defined
on R 2 0, 0, we know that g and h are continuous on R 2 0, 0. Note that h 0
on R 2 0, 0. Hence, f g/h is continuous on R 2 0, 0. Hence, f is continuous on
R 2 .
(d) By Exercise 4.11, we know that fx, y is continuous at 0, 0, where
x y sin1/x sin1/y if x 0andy 0,
fx, y
0 if x 0ory 0.
since lim x,y0,0 fx, y 0 f0, 0. It is the same method as in Exercise 4.11, we know
that f is discontinuous at x,0 for x 0andf is discontinuous at 0, y for y 0. And it is
clearly that f is continuous at x, y, wherex 0andy 0.
(e) By Exercise 4.11, Since
we rewrite
fx, y
fx, y
cos
sinxsiny
tan xtan y
,iftanx tan y,
cos 3 x if tan x tan y.
xy
cos xcos y
2
cos
xy
2
if tan x tan y
cos 3 x if tan x tan y.
We consider x, y /2, /2 /2, /2, others are similar. Consider two cases (1)
x y, and (2) x y, wehave
(1) (x y) Since lim x,ya,a fx, y cos 3 a fa, a. Hence, we know that f is
,
.