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Fixed point

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Hello, can someone please help me with the following problem:

Let f : [a, b] -> [a, b] be a homeomorphism such that f (a) = b and f (b) = a. Prove that f has exactly one fixed point.

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Solution Summary

This solution provides an example of proving a homeomorphism has exactly one fixed point.

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Proof:
Since f is a homeomorphism, then f has inverse g = f^(-1). Moreover, both f and g are continuous functions.
First, I claim that f has one fixed point.
We consider the function h(x) = f(x) - x. Since f is continuous, then h is also continuous. From the condition, we ...

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