The following analog of Kakutani's theorem has been proved by F. J. DYSON (MR44620)

Theorem (F. J. Dyson 1951): Let $\Bbb S^2$ be the surface of a sphere center $Z$ in Euclidean $3$-space $\Bbb R^3$, and let $f(x)$ be a continuous real-valued function defined on $\Bbb S^2$. Then there exist four points $x_1 , x_2 , x_3 , x_4$ on $\Bbb S^2$ forming the vertices of a square with center $Z$, such that $$f(x_1)=f(x_2)=f(x_3)=f(x_4).$$

Is the above theorem true for $\Bbb R^2=\Bbb S^2\backslash\{\infty\}$? i.e. there exist four points $x_1 , x_2 , x_3 , x_4$ on $\Bbb R^2$ forming the vertices of a square with center $Z$ (probably origin), such that $$f(x_1)=f(x_2)=f(x_3)=f(x_4)?$$

The following analogue of Kakutani's theorem has been proved by F. J. Dyson (MR44620)

Theorem (F. J. Dyson 1951): Let $\Bbb S^2$ be the surface of a sphere with center $Z$ in Euclidean $3$-space $\Bbb R^3$, and let $f(x)$ be a continuous real-valued function defined on $\Bbb S^2$. Then there exist four points $x_1 , x_2 , x_3 , x_4$ on $\Bbb S^2$ forming the vertices of a square with center $Z$, such that $$f(x_1)=f(x_2)=f(x_3)=f(x_4).$$

Is the above theorem true for $\Bbb R^2=\Bbb S^2\backslash\{\infty\}$? i.e. are there four points $x_1 , x_2 , x_3 , x_4$ on $\Bbb R^2$ forming the vertices of a square with center $Z$ (probably the origin), such that $$f(x_1)=f(x_2)=f(x_3)=f(x_4)?$$