# examples needed

• Oct 30th 2009, 03:55 PM
dannyboycurtis
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 $\displaystyle f:D\to \mathbb{R}$ with $\displaystyle D\subseteq \mathbb{R}$ closed and bounded which does not attain its maximum value in D.

Thanks in advance for any help!
• Oct 30th 2009, 05:39 PM
redsoxfan325
Quote:

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 $\displaystyle f:D\to \mathbb{R}$ with $\displaystyle D\subseteq \mathbb{R}$ 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 $\displaystyle \mathbb{R}$, because continuous functions on compact sets are bounded, so we have to be more adventurous.

Define a set $\displaystyle A=\{x\in\mathbb{Q}~|~2\leq x^2\leq3\}$ (This is basically the interval $\displaystyle [\sqrt{2},\sqrt{3}]$, but since we're in $\displaystyle \mathbb{Q}$, we have to be more careful when defining it.)

Then define the function $\displaystyle f:A\subset\mathbb{Q}\longrightarrow\mathbb{Q}$ given by $\displaystyle f(x)=\frac{1}{3-x^2}$

EDIT: Just noticed that only TWO of the conditions have to hold. In that case, take $\displaystyle f(x)=\frac{1}{x}$ on $\displaystyle (0,1]$.

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For the second one, the continuous function $\displaystyle f:(a,b)\subset\mathbb{R}\longrightarrow\mathbb{R}$ given by $\displaystyle f(x)=x$ will be bounded on $\displaystyle (a,b)$.

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$\displaystyle f:D=[0,2]\subset\mathbb{R}\longrightarrow\mathbb{R}$ given by:
$\displaystyle f(x)=\left\{\begin{array}{lr}x:&0\leq x<1\\0:&1\leq x\leq 2\end{array}\right\}$
$\displaystyle f$ has a $\displaystyle \sup$ in $\displaystyle D$, but no max.