Connectedness, Continuity, Image, Antipodal Point and Borsuk-Ulam Theorem

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).)

Please see the attached file for the fully formatted problems.

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