Show that, if X is a connected topological space and is continuous, then the image f(X) is an n interval.
Show that, if is a continuous map, then if given a,b,c in with a < b and c between f(a) and f(b), there exists at least one with a and f(x)=c
Let be a continuous map. Show that there exists a point in the circle such that f(x) =f(-x), where is the antipodal point of x. (hint: consider the function defined by g(x)=f(x)-f(-x).)
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Connectedness, continuity, image, antipodal point and borsuk-ulam theorem are investigated and discussed in the solution.