Can you do 24 please. Thanks 24. Suppose A is bounded and not compact. Prove that there is a function that is continuous onA, but not uniformly continuous. Give an example of a set that is not compact, but everyfunction continuous on that set is uniformly continuous.25. Give an example of sets A and B and a continuous function f: AUB→R such that f isuniformly continuous on A and uniformly continuous on B, but not uniformly continuous onAUB.*26. Let E C R. Prove that E is closed if, for every xo such that there is a sequence {x}=1 ofpoints of E converging to xo, it is true that xo E E. In other words, prove E is closed if itcontains all limits of sequences of members of E.*27. Prove that every set of the form (x: a < x < b} is open and every set of the form{x: a≤ x ≤ b}) is closed.28. Let D C R, and let D' be the set of accumulation points of D. Prove that D = DUD' isclosed and that if F is any closed set that contains D, then DC F. D is called the closureof D.29. If D CR is bounded, prove that D is bounded.

Let A be an open interval set, such that . This set is not compact since it is not closed. We have…

Suppose A is bounded and not compact. Prove that there is a function that is continuous on A, but not uniformly continuous. Give an example of a set that is not compact, but every function continuous on that set is uniformly continuous, Give an exam

Suppose A is bounded and not compact. Prove that there is a function that is continuous on A, but not uniformly continuous. Give an example of a set that is not compact, but every function continuous on that set is uniformly continuous, Give an exam

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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Question

Can you do 24 please. Thanks

24. Suppose A is bounded and not compact. Prove that there is a function that is continuous on
A, but not uniformly continuous. Give an example of a set that is not compact, but every
function continuous on that set is uniformly continuous.
25. Give an example of sets A and B and a continuous function f: AUB→R such that f is
uniformly continuous on A and uniformly continuous on B, but not uniformly continuous on
AUB.
*26. Let E C R. Prove that E is closed if, for every xo such that there is a sequence {x}=1 of
points of E converging to xo, it is true that xo E E. In other words, prove E is closed if it
contains all limits of sequences of members of E.
*27. Prove that every set of the form (x: a < x < b} is open and every set of the form
{x: a≤ x ≤ b}) is closed.
28. Let D C R, and let D' be the set of accumulation points of D. Prove that D = DUD' is
closed and that if F is any closed set that contains D, then DC F. D is called the closure
of D.
29. If D CR is bounded, prove that D is bounded.

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