An image of a sphere under continuous mapping. Let $\varphi:R^n\to R^n$ and $\varphi$ is continuous on the sphere $S=S(x,r)$. It is somehow clear that the image $\varphi(S)$ is a connected compact. My interest is what can we say about $R^n\setminus\varphi(S)$ ? I'm thinking in following way: if $\varphi$ is injective then $R^n\setminus\varphi(S)$ consist of two components bounded and unbounded. It is Jordan theorem in $R^n$. But I haven't found correct references for this assertion.

This is called the Jordan-Brouwer separation theorem. There is a sketch of proof in the Wikipedia article; the proof uses homology theoretical methods. A possible reference is this paper by Wolfgang Schmaltz (the proof is very much too complicated to be included here...).