Are All Real-Valued Functions with Zeros Zero Divisors?

[1800bigk]

Let R be the set of all real valued functions defined for all reals under function addition and multiplication.

Determine all zero divisors of R.

A zero divisor is a non zero element such that when multiplied with another nonzero element the product is zero.
So I said that the zero divisors of R would be all the functions in R that are not the zero function but take on the value of zero at least one time.
Am I close?
acoording to my answer though f=sin X and g= x-2 would be zero divisors because niether function is the zero function but fg = (sin X)(x-2) = 0 at x=2
but my teacher said this is wrong he said the product has to be zero for all x.
How would I even begin to find all the functions that are not the zero function but when multiplied together is zero for all x?

 

[HallsofIvy]

How are you defining multiplication in this case? Are you saying that if a function is 0 for any x then it is the "zero" function?

 

[1800bigk]

The problem says under normal function multiplication. The zero function is
Z(x)= 0 for all x

 

[HallsofIvy]

So f(x)= 2x is 0 for x= 0. But f*g is not, in general, equal to the 0 function.

 

[1800bigk]

so are there any zero divisors

 

[Hurkyl]

One thing I would do on a problem like this if I got stuck is to try and specialize the definitions to the situation at hand.

You know that f is a zero divisor iff there exists g such that f*g = 0.

So if f, g, and 0 are all functions, what form does this condition take?

 

[HallsofIvy]

What about, for example, f(x)= 0 if x<0, 1 if x>= 0; g(x)= 1 if x< 0; 0 if x>=0?

 

[1800bigk]

the piecewise functions above are both zero divisors. since neither function is the zero function but their product is zero for all x, ok I think I get now.

so any function that is not the zero function , but takes on the value of zero for some x, is a zero divisor. Becasue if my function f takes on zero for some interval I, I can always construct another function g that is zero on I compliment.

just to make sure I get this, f(x) = x^2 + 1 cannot be zero divisor because I would have to multiply it by a function g that's always zero, which would make g the zero function.

 

[HallsofIvy]

It doesn't have to be "intervals". Let f(x)= 0 if x is rational, 1 if x is irrational, g(x)= 1 if x is rational, 0 if x is irrational.

Your second statement is correct: if f(x) is NEVER 0 then it is not a zero divisor.

 

[Hurkyl]

Well, technically, if a function is zero anywhere, then it is zero on some interval (plus some other points).

 

[HallsofIvy]

Do you mean you are considering a singleton {a} as an interval?

 

[Hurkyl]

Sure, since {a} = [a, a]