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    Continuous function on a circle and antipodal points

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    Let S be the circle in the plane {(x,y) in R^2: x^2 + y^2 = 1} and f: S --> R be a continuous map. Show that there exists (x,y) on S such that f(x,y) = f(-x,-y).

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    The idea is to transform the problem into a one variable problem. One way is to use complex variables where the circle is {e^{i t}:0<= t<= 2pi} and another is to just solve for y. We do the latter. The upper half of the circle is the graph of the function ...

    Solution Summary

    We transform the problem into a one-variable calculus problem. We use the intermediate value theorem to solve this question.