Paper 1, Section II, E

Analysis I | Part IA, 2011

State and prove the Intermediate Value Theorem.

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

Answer the following questions with justification.

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

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

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

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

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