2. Let f : R → R be a continuous function. Suppose that (f o f) (x) = x for every x E R. (a) Prove that there exists a point c e R such that f (c) = c. Hint: Consider the function g (x) = f (x) – x for every x E R. (b) Give an example for a function satisfying the above requirements, and such that f # id, -id , where id is the identity function.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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2. Let f : R → R be a continuous function. Suppose that (f o f) (x) = x for
every x E R.
(a) Prove that there exists a point c ER such that f (c)
Hint: Consider the function g (x) = f (x) – x for every x € R.
= C.
(b) Give an example for a function satisfying the above requirements,
and such that f id, -id , where id is the identity function.
Transcribed Image Text:2. Let f : R → R be a continuous function. Suppose that (f o f) (x) = x for every x E R. (a) Prove that there exists a point c ER such that f (c) Hint: Consider the function g (x) = f (x) – x for every x € R. = C. (b) Give an example for a function satisfying the above requirements, and such that f id, -id , where id is the identity function.
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