Paper 1, Section II, E

State and prove the Intermediate Value Theorem.

A fixed point of a function $f: X \rightarrow X$ is an $x \in X$ with $f(x)=x$. Prove that every continuous function $f:[0,1] \rightarrow[0,1]$ has a fixed point.

Answer the following questions with justification.

(i) Does every continuous function $f:(0,1) \rightarrow(0,1)$ have a fixed point?

(ii) Does every continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ have a fixed point?

(iii) Does every function $f:[0,1] \rightarrow[0,1]$ (not necessarily continuous) have a fixed point?

(iv) Let $f:[0,1] \rightarrow[0,1]$ be a continuous function with $f(0)=1$ and $f(1)=0$. Can $f$ have exactly two fixed points?

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