Let f : R → R be a continuous function satisfying (fof)(x) = x Vx € R. a) Show that f has a fixed point. b) Prove: If f is monotonically increasing, then f (x) = x for all x E R.

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Let f : R → R be a continuous function satisfying
(fof)(x) = x
Vx e R.
a) Show that f has a fixed point.
b) Prove: If f is monotonically increasing, then f (x) = x for all x € R.
Transcribed Image Text:Let f : R → R be a continuous function satisfying (fof)(x) = x Vx e R. a) Show that f has a fixed point. b) Prove: If f is monotonically increasing, then f (x) = x for all x € R.
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