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Consider the two dimensional real place $\mathbb{R}^2$. We have a triangle (lets call it $\triangle$) with the vertices $(\frac{\sqrt{3}}{2},\frac{3}{2}),~(-\frac{\sqrt{3}}{2},\frac{3}{2})$ and $(0,-\sqrt{3})$. I need to find ALL continuous maps $f:\mathbb{R}^2\longrightarrow \mathbb{R}^2$ such that $f(\triangle)\hookrightarrow\triangle$.

I tried to convert $(x,y)\in\mathbb{R}^2$ as complex number $z=x+\imath y$ and tried to calculate the analytic ones as well. However, it is becoming too complicated at the end. Can someone please suggest some elegant method to handle the above problems (or at the least, determine all analytic $f$ with above properties). Advanced thanks for any reply.

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wait when you say ALL maps $f$, do have in mind set-theoretic maps, or continuous maps, or real analytic maps, or smooth maps, etc. ? –  uncookedfalcon Aug 23 '12 at 21:16
    
@uncookedfalcon actually i wanted to get all continuous maps. however it is also nice to get any information about (real) analytic maps (or complex one which i tried to write from möbius transformation, as well. thanks for the comment. edited the question. –  BIrendra Sen Aug 24 '12 at 3:50
    
There is a lot of difference between continuous and complex analytic, that you essentially posed two separate problems. Concerning the analytic case, does the map have to be a bijection between triangles? –  user31373 Aug 24 '12 at 4:49
    
@BIrendraSen I strongly doubt that there is a reasonable way to describe the set of all continuous such maps. If you just want the image to be included in the target triangle, I also doubt there is a way to describe the analytic ones. On second thought: What do you mean by a triangle? Just the edges or the interior too? If it's just the edges you're talking about, the problem seems a lot more feasible. –  mrf Aug 24 '12 at 8:58
    
@mrf actually by the word triangle, i wanted to mean all the points inside (and upon) the triangle, i.e. the closed set. –  BIrendra Sen Aug 26 '12 at 17:50
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