# Borsuk-Ulam theorem and two different pairs of antipodal points

Borsuk - Ulam theorem states in particular that for any continous map $$f:S^2\rightarrow \mathbb{R}^{2}$$, there exists point $$x\in S^{2}$$ such that $$f(-x)=f(x)$$. I call $$(x,-x)$$ a "good" pair

My question is:

How should $$f$$ look like if we want to have at least two different good pairs? This is $$x,y\in S^2$$ such that $$f(x)=f(-x)$$ and $$f(y)=f(-y)$$ Is the answer connected to differentiability of $$f$$?

More generally i am intrested in the structure of set of these antipodal points. When does this set contain continuum of good pairs?

Regards

Obtaining a structure theorem on the set of good pairs for all such continuous functions is at best impossible; however, some specialized examples can answer the other questions. Trivially, a constant function gives you a continuum of good pairs. Regarding differentiable function with a single good pair, take for example $$f(x,y,z):=(x,y)$$ where $$x^2+y^2+z^2=1$$ and observe that the only good pair is the north and south poles —that is, $$(0,0,1)$$ and $$(0,0,-1)$$.