Thread: examples needed

Hi y'all I was just wondering if anyone could help me out with some examples of the following:
a function f where two of the following conditions hold:
--f is continuous on [a,b],
--the interval [a,b] is closed,
--the interval [a,b] is bounded,
yet f is not bounded on [a,b].

a function f where two of the previous conditions hold and f IS bounded on [a,b].

a function f where \to \mathbb{R}" alt="f\to \mathbb{R}" /> with closed and bounded which does not attain its maximum value in D.

Thanks in advance for any help!

Originally Posted by dannyboycurtis

Hi y'all I was just wondering if anyone could help me out with some examples of the following:
a function f where two of the following conditions hold:
--f is continuous on [a,b],
--the interval [a,b] is closed,
--the interval [a,b] is bounded,
yet f is not bounded on [a,b].

a function f where two of the previous conditions hold and f IS bounded on [a,b].

a function f where \to \mathbb{R}" alt="f\to \mathbb{R}" /> with closed and bounded which does not attain its maximum value in D.

Thanks in advance for any help!

Well for the first one, it can't be a function in , because continuous functions on compact sets are bounded, so we have to be more adventurous.

Define a set (This is basically the interval , but since we're in , we have to be more careful when defining it.)

Then define the function given by

EDIT: Just noticed that only TWO of the conditions have to hold. In that case, take on .

--------

For the second one, the continuous function a,b)\subset\mathbb{R}\longrightarrow\mathbb{R}" alt="fa,b)\subset\mathbb{R}\longrightarrow\mathbb{R}" /> given by will be bounded on .

----------

I'm not quite sure I understand what the third question is asking, but what about this:

=[0,2]\subset\mathbb{R}\longrightarrow\mathbb{R}" alt="f=[0,2]\subset\mathbb{R}\longrightarrow\mathbb{R}" /> given by:



has a in , but no max.