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While proving, that given two bounded polygonal region in $R^2$, there exists a line in $R^2$ that bisects each of them (the bisection theorem), munkres progresses with taking the the polygonal regions in XY plane translated by unit distance along the positive z-axis and then he considers points u in $S^2$ and by constructing a plane which has u as its unit normal vector and passing through the origin. The plane clearly divides $R^2$ in two half-spaces, if we let $f_i(u)$ be the area of the portion of $A_i$ that lies on same side of P as does the vector u, then why is the map $ u \rightarrow f_1(u) $ continuous?

Intuitively the picture is clear, as we move u continuously along the sphere the intersecting line between the two planes also moves continuously and the area changes continuously too.

But writing out a formal proof is turning out to be difficult for me, and thanks for the suggestions and answers in advance.

Once we prove that the rest will follow from borsuk-ulam theorem

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