Vtyjkyuk

Question. Let $C(BbbR)$ denote the ring of real-valued continuous functions on $BbbR$, with pointwise addition and multiplication. Which of the following form an ideal in this ring?

1. The set of all $C^infty$ functions having compact support.


2. $C_c(BbbR)$


3. The set of all continuous functions which vanish at infinity.

My Solution.

2. True.(Follows from: $Support(f+g) subset support(f) cup support(g)$ also $Support(fg) subset support(f)$ and $support(f):=cl f(x) neq 0$)




3. False. (Take, $f(x)=frac11+x^2 in C_0(BbbR)$ and $r(x)=1+x^2in C(BbbR)$, then $rf$ doesn't belong to $C_0(BbbR)$).

But I cannot find a counter example in (1) as I did in (3) ...

Suppose I choose $f(x)=1$ in $[-1,1]$ and take $r(x)=|x|$. Then $rf(x)=|x|$ and $rf$ doesn't belong to $C_0(BbbR)$. But the problem is if I define $f=0$ outside $[-1,1]$ it wouldn't be smooth...

So I think I have to find another counter example. Can any one please help me to find an counter example here?

There exists a $C^infty$ function $f$ such that $f(x)=1$ for all $x in (-1,1)$ and $f(x)=0$ for $|x| >2$. [Construction of such functions using $e^-1/x$ is standard]. If you multiply this by $|x|$ you will go out of $C^infty$.