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1. Suppose f is a function which is differentiable in an open interval I, that
|f'(x)|<=M<1 for all x e I and that f(alpha) = alpha for some alpha e I. Show that there is no other solution of f(x) = x in the interval I.
2. Suppose b is another solution so that f(b)=b. Now the mean value theorem tells us:
(f(b) - f(a))/(b-a) = f'(c), for some c between a and b

But f(b) - f(a) = b - a, and so f'(c) = 1, but that's impossible as |f(x)|<1. Hence there can be no other solution.
3. (Original post by Zagani)
Suppose f is a function which is differentiable in an open interval I, that
|f'(x)|<=M<1 for all x e I and that f(alpha) = alpha for some alpha e I. Show that there is no other solution of f(x) = x in the interval I.
Hold on - I answered an identical question on here yesterday. It's not some kind of assessed coursework/homework job is it?

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Updated: February 27, 2006
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