Let $$ S_1=\{(x,y)\in\mathbb{R^2}:1<x^2+y^2<2\} $$ $$ S_2=\{(x,y)\in\mathbb{R^2}:2x^2<x^2+y^2<4\} $$ Is there a continuous function $f$ mapping $\ S_1\ $ onto $\ S_2\ $?
I believe that there is no function $f$ that maps $\ S_1\ $ onto $\ S_2\ $.
My reason for this is that while $S_1$ is path connected, $S_2$ is not path connected and $S_1\cap S_2$ is also not path connected. Hence you cannot map $S_1$ onto $S_2$. Is this logic correct? How can I build on this explanation?
You are right. The set $S_1$ is path-connected. Therefore, if $f$ is a continuous function whose domain is $S_1$, $f(S_1)$ is path-connected too. But $S_2$ is not path-connected, since $(0,\pm1)\in S_2$, but there is no element $(x,y)\in S_2$ such that $y=0$.
You can use the same argument using connectedness instead of path-connectedness.