1. subrings

Are the following subrings of the ring of all functions from the closed interval $[0,1]$ to $\mathbb{R}$:
1. the set of all functions which have only a finite number of zeros, together with the zero function
2. the set of all functions $f$ such that $lim_{x -> 1^{-}} f(x) = 0$

2. Originally Posted by dori1123
Are the following subrings of the ring of all functions from the closed interval $[0,1]$ to $\mathbb{R}$:
1. the set of all functions which have only a finite number of zeros, together with the zero function
2. the set of all functions $f$ such that $lim_{x -> 1^{-}} f(x) = 0$
For the first one, think about what kind of functions have finite zeros or are constant zero. The polynomials form a ring, but are there other elements in this subring?

For the second, simply check the subring properties. If you add two functions with this limit property, does the result also have this limit? It is straightforward.

3. Originally Posted by robeuler
The polynomials form a ring, but are there other elements in this subring?
Yes there are many because there is no condition on continuity.

4. Originally Posted by robeuler
For the first one, think about what kind of functions have finite zeros or are constant zero.
A function with finite zeros, does that mean finite number of roots?
I am trying to show it's closed under multiplication, so the product of two functions with finite number of zeros will also have finite number of zeros. But how can I show that?

5. Ok you seem to be heading in the wrong direction because the first one is not a subring.

Here is a counter example:
$f(x)=1$
and
$g(x)=\left\{\begin{array}{cc}-1,&\mbox{ if }x\leq 0.5\\1,&\mbox{ if }x>0.5\end{array}\right.$
Both f and g are in the subring but
$f(x)+g(x)=\left\{\begin{array}{cc}0,&\mbox{ if }
x\leq 0.5\\2, & \mbox{ if } x>0.5\end{array}\right.$

is not.

6. Why isn't $f(x)+g(x)=\left\{\begin{array}{cc}0,&\mbox{ if }
x\leq 0.5\\2, & \mbox{ if } x>0.5\end{array}\right.$
in the set?

7. There is an infinite number of zeroes because there is an infinite number of x values between 0 and 0.5.

8. Thanks, I got it.