Math Is Fun Forum

alice8675309

Member

Registered: 2010-04-06

Posts: 2

Examples of continuous functions who are bounded and unbounded??

Let a,b be real numbers with a < b.
(1) Give an example of a continuous f on [a,b) which is not bounded.
(2) Give an example of a continuous f on [a,infinity) which is not bounded.
(3) Give an example of a bounded f on [a,b] for which sup_[a,b] f is not achieved.
(4) Give an example of a bounded, continuous f on [a,infinity) for which sup_[a,infinity) f is not achieved.
(5) Give an example of a bounded, continuous f on [a,b) for which sup_[a,b) f is not achieved.

I just get so confused, I feel like i need to see the examples to understand them.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Examples of continuous functions who are bounded and unbounded??

(1) Give an example of a continuous f on [a,b) which is not bounded.

A way to think about this question is to name a function which goes off to infinity at some number.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Examples of continuous functions who are bounded and unbounded??

Let a,b be real numbers with a < b.
(1) Give an example of a continuous f on [a,b) which is not bounded.
(2) Give an example of a continuous f on [a,infinity) which is not bounded.
(3) Give an example of a bounded f on [a,b] for which sup_[a,b] f is not achieved.
(4) Give an example of a bounded, continuous f on [a,infinity) for which sup_[a,infinity) f is not achieved.
(5) Give an example of a bounded, continuous f on [a,b) for which sup_[a,b) f is not achieved.

I just get so confused, I feel like i need to see the examples to understand them.

I’m not doing your homework for you but I’ll show you examples using intervals with 0 and 1 as end points so as to give you the picture of what’s required. (Often examples explain things better than wordy definitions.)

If you get the idea, do the problems with a and b instead of 0 and 1.

Last edited by JaneFairfax (2010-04-07 00:59:15)