Bounded function

**mus** · January 13th 2011, 12:00 PM

Can someone assist me in the following:

Let V be the space of continuous functions $f: \mathbb{R} \to \mathbb{R}$ , such that:

$\forall \epsilon >0 \exists a\in \mathbb{R}^{+} \forall x \in \mathbb{R}: |x|>a \Rightarrow |f(x)|< \epsilon$

I have shown that $V$ is a vector space.

How do I show that every $f \in V$ is bounded.

**Drexel28** · January 13th 2011, 12:12 PM

Originally Posted by mus

Can someone assist me in the following:

Let V be the space of continuous functions $f: \mathbb{R} \to \mathbb{R}$ , such that:

$\forall \epsilon >0 \exists a\in \mathbb{R}^{+} \forall x \in \mathbb{R}: |x|>a \Rightarrow |f(x)|< \epsilon$

I have shown that $V$ is a vector space.

How do I show that every $f \in V$ is bounded.

These are called functions of compact support. They're obviously bounded since there exists some compact set $C\subseteq\mathbb{R}$ such that $\displaystyle \sup_{x\in\mathbb{R}-C}|f(x)|<1$ and so for every $x\in\mathbb{R}$ one has $\displaystyle |f(x)|\leqslant \max\{1,\sup_{x\in C}|f(x)|\}$ . So answer me this why is $\displaystyle \sup_{x\in C}|f(x)|<\infty$ .

**Jose27** · January 13th 2011, 12:38 PM

Originally Posted by Drexel28

These are called functions of compact support.

Actually, functions with compact support are those which are zero outside of some compact set. This space V (sometimes denoted $C_0(\mathbb{R})$ ) is the space of functions vanishing at infinity.

**surjective** · January 13th 2011, 01:14 PM

Could you kindly elaborate on the above. It is too abstract!!

**Plato** · January 13th 2011, 01:47 PM

Originally Posted by mus

Let V be the space of continuous functions $f: \mathbb{R} \to \mathbb{R}$ , such that:
$\forall \epsilon >0 \exists a\in \mathbb{R}^{+} \forall x \in \mathbb{R}: |x|>a \Rightarrow |f(x)|< \epsilon$
How do I show that every $f \in V$ is bounded.

In the above let $\epsilon =1$ then there is $a_1>0$ such that $|x|>a_1\Rightarrow ~|f(x)|< 1$ .
Now $f$ is bounded on $[-a_1,a_1]$ . WHY?
Does that mean that $f$ is bounded on $\mathbb{R}~?$

**Drexel28** · January 13th 2011, 02:17 PM

Originally Posted by Jose27

Actually, functions with compact support are those which are zero outside of some compact set. This space V (sometimes denoted $C_0(\mathbb{R})$ ) is the space of functions vanishing at infinity.

Of course. Stupid slip of the tongue.

**mus** · January 13th 2011, 03:49 PM

Don't see how f it is bounded on [-a1;a1]? Please explain!

**Plato** · January 13th 2011, 04:05 PM

Originally Posted by mus

Don't see how f it is bounded on [-a1;a1]? Please explain!

Are saying that you are working on the sorts of problems and do not know that continuous functions on finite closed intervals are bounded? If that is true, then you need to sit down a live tutor and work through you misunderstandings.

**mus** · January 13th 2011, 04:14 PM

This subject is completely new to me and when your teacher rushes through the book without giving time to study the theorems then what do you expect? By the way I did not post to get talked down to but to be enlightened. So if I lack the basics then either my tutor could give that to me or you could be helpful enough to do so. I could say alot more but I will refrain. Could anyone else be kind enough to help me through this problem without the sarcasm.

Thanks a lot.

**Plato** · January 13th 2011, 04:24 PM

Originally Posted by mus

This subject is completely new to me and when your teacher rushes through the book without giving time to study the theorems then what do you expect? By the way I did not post to get talked down to but to be enlightened. So if I lack the basics then either my tutor could give that to me or you could be helpful enough to do so. I could say alot more but I will refrain. Could anyone else be kind enough to help me through this problem without the sarcasm

There in nothing in my reply to suggest that I meant to make you feel bad. I simply suggested that you have a one-on one, face to face contact with you instructor.
It is simply a fact that: Continuous functions on compact sets are bounded. If you do not understand that, then you need ‘live’ help.

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