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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).

https://brainmass.com/math/complex-analysis/continuous-function-circle-antipodal-points-515573

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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.

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