Q: Changing arrangement of square's vertices with a bijective continuous map in plane. Is there a continuous bijective map $S:\mathbb{R^2}\to \mathbb{R^2}$ which convert vertices of Square $ABCD$ (respectly arranged points) to vertices of Square $A'C'B'D'$ ? 
(each points goes to its prime)

A: In order to simple the question we just need to replace $B$ and $C$ and let $A$ and $D$ fixed. Now consider an annuli contains $B$ and $C$ on two side of a diameter of mid circle of the annuli,the map rotates points on each concentric circles of the annuli by gradually increasing angles between $(0,2\pi)$ when radius of the circles goes bigger, eventually $B$ and $C$ which lie on the mid circle of the annuli rotates $\pi$ and substitute eachother.here we just moved points in the annuli and the other points remains fixed.

A: I am sure there is a simpler argument, but you can use the isotopy extension theorem for this. Find some isotopy that moves the given points in the plane (in this case $A,B,C,D$) to the new points you want.  This then extends to an ambient isotopy of the plane, and in particular gives you a homeomorphism. 
