I was trying to map a set x=(0,n) to y=(0,1) by using a single(non-piecewise) function.
For example, you could say:
f(x)={0, x=0 && 1, x \dne 0}
however this is a piecewise function.
I wanted the function to like this f(x) = x/x. However, this leaves x=0 as undefined.
I have tried other ways but it always seems that you must divide by some f(x) which could have the possibility of being 0.
Thus, I am trying to prove that:
(0,n) can't map to (0,1) without dividing by some f(n) where f(n)=0.
Has this been proven? Or, how would I go about proving it?
Let $n = 1$. Then every element maps to itself.
If $n \neq 1$, then let $f(x) = x/n : x \in (0, n)$.
$0$ is not in the domain of $n$ because $0$ cannot map to $(0,1)$ surjectively.