Limits of Functions in Real Analysis

**zebra2147** · October 19th 2010, 05:51 PM

Alright, I need some guidance with the following problem. Any help would be appreciated.
Suppose $lim_{x\rightarrow x_{0}}f(x)=b$ and given any $r>0$ the set $f(B'_{r}(a))$ contains both poisitive and negative real numbers,show that $b=0$ .

After I talked to some people it sounds like there may be a typo???
Possibly $x_{0}=a$ in this problem? Either way I'm not really sure how to show that b=0.

**tonio** · October 19th 2010, 07:28 PM

Originally Posted by zebra2147

Alright, I need some guidance with the following problem. Any help would be appreciated.
Suppose $lim_{x\rightarrow x_{0}}f(x)=b$ and given any $r>0$ the set $f(B'_{r}(a))$ contains both poisitive and negative real numbers,show that $b=0$ .

After I talked to some people it sounds like there may be a typo???
Possibly $x_{0}=a$ in this problem? Either way I'm not really sure how to show that b=0.

It must be, again, $a=x_0$ : just think that when $x\rightarrow x_0$ , x gets into ANY neighborhood of $x_0$ , so if in ANY such

neighborhood there are positive and negative values of f AND the limit exists, this limit must be zero.

Try now to give a proof with $\epsilon\,,\,\delta$ and/or a proof bya contradiction.

Tonio

**zebra2147** · October 20th 2010, 05:13 AM

Attempt #1 at proof:

Suppose $D=D_{1}\bigcup D_{2}$ where $D_{1}$ contains only negative real numbers and $D_{2}$ contains only positive real values. Let $a$ be a limit point of $D_{1}$ and $D_{2}$ . Then let $f: D\rightarrow R$ be the function $f(B'_{r}(a))$ . Thus, the $lim_{x\longrightarrow a}$ exists only if $lim_{x\longrightarrow a}f|_D_{j}$ both exist and are equal.

Then, if both limits exist we have that, for some $\epsilon>0$ ,when $x_{1}<0$ , $|x_{1}-a|<\delta\Rightarrow |f(B'_{r}(a))-b|<\epsilon$ . Also, when $x_{2}>0$ , $|x_{2}-a|<\delta\Rightarrow |f(B'_{r}(a))-b|<\epsilon$ .

The only way that this can happen is when $b=0$ ?????

**tonio** · October 20th 2010, 09:24 PM

Originally Posted by zebra2147

Attempt #1 at proof:

Suppose $D=D_{1}\bigcup D_{2}$ where $D_{1}$ contains only negative real numbers and $D_{2}$ contains only positive real values. Let $a$ be a limit point of $D_{1}$ and $D_{2}$ . Then let $f: D\rightarrow R$ be the function $f(B'_{r}(a))$ . Thus, the $lim_{x\longrightarrow a}$ exists only if $lim_{x\longrightarrow a}f|_D_{j}$ both exist and are equal.

Then, if both limits exist we have that, for some $\epsilon>0$ ,when $x_{1}<0$ , $|x_{1}-a|<\delta\Rightarrow |f(B'_{r}(a))-b|<\epsilon$ . Also, when $x_{2}>0$ , $|x_{2}-a|<\delta\Rightarrow |f(B'_{r}(a))-b|<\epsilon$ .

The only way that this can happen is when $b=0$ ?????

I think you confused stuff here big time: the positive and negative numbers are in the image of f , NOT in its domain!

We know that $\forall\epsilon>0\,\exists\delta_\epsilon>0\,\,s.t .\,\,x\in B_{\delta_\epsilon}(x_0)\Longrightarrow f(x)\in B_\epsilon(b)$ .

Suppose $b>0$ , and let us choose $\epsilon=b/2$ . By the above, we get that $f(x)\in B_\epsilon(b)$ for any

point $x\in B_{\delta_\epsilon}(x_0)$ , for some $\delta_\epsilon>0$ ,or in other words: $f\left(B_{\delta_\epsilon}(x_0)\right)\subset B_\epsilon(b)$ ...but this can't be since

in $f\left(B_{\delta_\epsilon}(x_0)\right)$ there's some negative number, whereas in $B_\epsilon(b)$ all the numbers are positive...

Read the above slowly and with a pencil and a piece of paper by your side...draw diagrams if that helps you.

Tonio

**zebra2147** · October 21st 2010, 05:11 AM

Oh ok. That makes more sense. Then I'm assuming that you showed how b>0 does not work, so if we show that b<0 does not work by a similar proof then b=0 must be true?

**tonio** · October 21st 2010, 08:02 PM

Originally Posted by zebra2147

Oh ok. That makes more sense. Then I'm assuming that you showed how b>0 does not work, so if we show that b<0 does not work by a similar proof then b=0 must be true?

Of course. I think you don't even need to show for $b<0$ since it is plain that this case is completely similar to the one already proven.

Tonio

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