1. Show that f(x) = x3 is not uniformly continuous on R. 2. Show that f(x) = 1/(x-2) is not uniformly continuous on (2,00). 3. Show that f(x)=sin(1/x) is not uniformly continuous on (0,л/2]. 4. Show that f(x) = mx + b is uniformly continuous on R. 5. Show that f(x) = 1/x2 is uniformly continuous on [1, 00), but not on (0, 1]. 6. Show that if f is uniformly continuous on [a, b] and uniformly continuous on D (where D is either [b, c] or [b, 00)), then f is uniformly continuous on [a, b]U D. 7. Show that f(x)=√x is uniformly continuous on [1, 00). Use Exercise 6 to conclude that f is uniformly continuous on [0, ∞). 8. Show that if D is bounded and f is uniformly continuous on D, then fis bounded on D. 9. Let f and g be uniformly continuous on D. Show that f+g is uniformly continuous on D. Show, by example, that fg need not be uniformly con- tinuous on D. 10. Complete the proof of Theorem 4.7. 11. Give an example of a continuous function on Q that cannot be continuously extended to R. 12. Let f : Q→ R be uniformly continuous on Q. Show that f has a unique continuous extension to R. [Hint: Define g: RR by [f(x) g(x) = lim f(x) 818 if x = Q if xe R\Q and (xn)nen is a sequence in Qwith xn → x. 818 First show that g is well-defined; that is, lim f(x,) is a real number and if (yn)nen is another sequence in Q with ynx, then lim f(x): lim f(y). Then show directly (e-8 argument) that g is uniformly con- 818 tinuous on R 1 818 =
1. Show that f(x) = x3 is not uniformly continuous on R. 2. Show that f(x) = 1/(x-2) is not uniformly continuous on (2,00). 3. Show that f(x)=sin(1/x) is not uniformly continuous on (0,л/2]. 4. Show that f(x) = mx + b is uniformly continuous on R. 5. Show that f(x) = 1/x2 is uniformly continuous on [1, 00), but not on (0, 1]. 6. Show that if f is uniformly continuous on [a, b] and uniformly continuous on D (where D is either [b, c] or [b, 00)), then f is uniformly continuous on [a, b]U D. 7. Show that f(x)=√x is uniformly continuous on [1, 00). Use Exercise 6 to conclude that f is uniformly continuous on [0, ∞). 8. Show that if D is bounded and f is uniformly continuous on D, then fis bounded on D. 9. Let f and g be uniformly continuous on D. Show that f+g is uniformly continuous on D. Show, by example, that fg need not be uniformly con- tinuous on D. 10. Complete the proof of Theorem 4.7. 11. Give an example of a continuous function on Q that cannot be continuously extended to R. 12. Let f : Q→ R be uniformly continuous on Q. Show that f has a unique continuous extension to R. [Hint: Define g: RR by [f(x) g(x) = lim f(x) 818 if x = Q if xe R\Q and (xn)nen is a sequence in Qwith xn → x. 818 First show that g is well-defined; that is, lim f(x,) is a real number and if (yn)nen is another sequence in Q with ynx, then lim f(x): lim f(y). Then show directly (e-8 argument) that g is uniformly con- 818 tinuous on R 1 818 =
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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![1. Show that f(x) = x3 is not uniformly continuous on R.
2. Show that f(x) = 1/(x-2) is not uniformly continuous on (2,00).
3. Show that f(x)=sin(1/x) is not uniformly continuous on (0,л/2].
4. Show that f(x) = mx + b is uniformly continuous on R.
5. Show that f(x) = 1/x2 is uniformly continuous on [1, 00), but not on
(0, 1].
6. Show that if f is uniformly continuous on [a, b] and uniformly continuous
on D (where D is either [b, c] or [b, 00)), then f is uniformly continuous
on [a, b]U D.
7. Show that f(x)=√x is uniformly continuous on [1, 00). Use Exercise 6
to conclude that f is uniformly continuous on [0, ∞).
8. Show that if D is bounded and f is uniformly continuous on D, then fis
bounded on D.
9. Let f and g be uniformly continuous on D. Show that f+g is uniformly
continuous on D. Show, by example, that fg need not be uniformly con-
tinuous on D.
10. Complete the proof of Theorem 4.7.
11. Give an example of a continuous function on Q that cannot be continuously
extended to R.
12. Let f : Q→ R be uniformly continuous on Q. Show that f has a unique
continuous extension to R. [Hint: Define g: RR by
[f(x)
g(x) = lim f(x)
818
if x = Q
if xe R\Q and (xn)nen is a
sequence in Qwith xn → x.
818
First show that g is well-defined; that is, lim f(x,) is a real number and
if (yn)nen is another sequence in Q with ynx, then lim f(x):
lim f(y). Then show directly (e-8 argument) that g is uniformly con-
818
tinuous on R 1
818
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f89dc84-5360-495d-a77a-10dc4ac0ca49%2F49fe565b-41e8-4e89-8bb0-d071b1b9f042%2Fbk1ij9b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Show that f(x) = x3 is not uniformly continuous on R.
2. Show that f(x) = 1/(x-2) is not uniformly continuous on (2,00).
3. Show that f(x)=sin(1/x) is not uniformly continuous on (0,л/2].
4. Show that f(x) = mx + b is uniformly continuous on R.
5. Show that f(x) = 1/x2 is uniformly continuous on [1, 00), but not on
(0, 1].
6. Show that if f is uniformly continuous on [a, b] and uniformly continuous
on D (where D is either [b, c] or [b, 00)), then f is uniformly continuous
on [a, b]U D.
7. Show that f(x)=√x is uniformly continuous on [1, 00). Use Exercise 6
to conclude that f is uniformly continuous on [0, ∞).
8. Show that if D is bounded and f is uniformly continuous on D, then fis
bounded on D.
9. Let f and g be uniformly continuous on D. Show that f+g is uniformly
continuous on D. Show, by example, that fg need not be uniformly con-
tinuous on D.
10. Complete the proof of Theorem 4.7.
11. Give an example of a continuous function on Q that cannot be continuously
extended to R.
12. Let f : Q→ R be uniformly continuous on Q. Show that f has a unique
continuous extension to R. [Hint: Define g: RR by
[f(x)
g(x) = lim f(x)
818
if x = Q
if xe R\Q and (xn)nen is a
sequence in Qwith xn → x.
818
First show that g is well-defined; that is, lim f(x,) is a real number and
if (yn)nen is another sequence in Q with ynx, then lim f(x):
lim f(y). Then show directly (e-8 argument) that g is uniformly con-
818
tinuous on R 1
818
=
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