Let $R$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside $(-1,1)$ and let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside a bounded open interval containing zero (depended on $f$). Is it true that $R \cong S$?



Are these rings of functions isomorphic?
They are not ring isomorphic, because e.g. $R$ has the following property of a ring $X$, and $S$ does not:



> 
> There is a non zero element $u \in X$ such that for any
>  invertible $f\in X$ either $uf$ or $-uf$ is a square, and for some $g\in X$ 
>  neither $ug$ nor $-ug$ is a square.
> 
> 
> 


