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Continuous function on compact space

Show that if f is a continuous real-valued function on the compact space X,
then there exist points x_1, x_2 in X such that
f(x_1)=inf{f(x):x in X} and
f(x_2)=sup{f(x):x in X}.

Solution Summary

The solution contains the proof of the theorem "if f is a continuous real-valued function on the compact space X,
then there exist points x_1, x_2 in X such that
f(x_1)=inf{f(x):x in X} and
f(x_2)=sup{f(x):x in X}".

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