Analysis by Ekkehard Kopp

Author:Ekkehard Kopp [Kopp, P E]
Language: eng
Format: epub, pdf
ISBN: 978-0-08-092872-2
Publisher: Elsevier Science
Published: 1996-04-18T04:00:00+00:00

7.2 Applications: fixed points, roots and iteration

Having worked hard to establish our theorems we can now collect some of the results:

Proposition 2: Let be a continuous function. Then f has a fixed point in [a, b]; that is, there is at least one c ∈ [a, b] such that f (c) = c.

PROOF: We apply the IVT to the continuous function g defined by g(x) = f (x) − x. Since the range of f is contained in [a, b], we must have f (a) ≥ a and f (b) ≤ b. Hence , while . By the IVT there exists c ∈ [a, b] such that g(c) = 0, i.e. f (c) = c.

This simple consequence of the IVT turns out to be quite difficult to generalize to higher dimensions – though it remains true in much greater generality.

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