Exercise 9.7.2. Let f : [0,1]
[0,1] be a continuous function. Show that
there exists a real number x in [0,1] such that f(x) = x. (Hint: apply the
intermediate value theorem to the function f(x) - x.) This point x is known
as a fixed point of f, and this result is a basic example of a fixed point theorem,
which play an important rôle in certain types of analysis.
Theorem 9.7.1 (Intermediate value theorem). Let a < b, and let f :
R be a continuous function on [a,b]. Let y be a real number
between f(a) and f (b), i.e., either f(a) < y < f (b) or f (a) >y > f (b) .
Then there exists CE [a, b] such that f (c) = y.
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