Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
![(3) Suppose that ƒ : [0, 1] → R is a real function. with ƒ([0, 1]) ≤ [0, 1]. Show that there
exists an a € [0, 1] with f(a) = a. (Hint: Consider the function g(x) = x − f(x).)](https://content.bartleby.com/qna-images/question/86c8dcbb-d46d-4c91-a740-ef32ebf33ae0/8e238125-ee1b-4dfb-8609-7127742e5205/cjsozi_processed.png)
Transcribed Image Text:(3) Suppose that ƒ : [0, 1] → R is a real function. with ƒ([0, 1]) ≤ [0, 1]. Show that there
exists an a € [0, 1] with f(a) = a. (Hint: Consider the function g(x) = x − f(x).)
Expert Solution
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Step 1
Sol:-
Let g(x) = x - f(x). Then g(x) is also a function from [0, 1] to R.
Notice that
g(0) = 0 - f(0) = -f(0)
and
g(1) = 1 - f(1) = 1 - f(1),
and
g([0, 1]) = [0 - f(0), 1 - f(1)] = [-f(0), 1 - f(1)] ⊆ [0, 1].
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