If a function f is uniformly continuous on R and f(x^2) = f(x) for all x, then prove that f is a constant function.

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Asked 11/17/2010 10:10:21 PM

Updated 6/4/2023 12:10:26 AM

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Asked 11/17/2010 10:10:21 PM

Updated 6/4/2023 12:10:26 AM

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Assume f is not a constant function. By contradiction, for any > 0, there exists > 0 such that |f(x^2) - f(x)| < for all x. However, choosing = |f(a) - f(b)|, where a and b are distinct, leads to a contradiction. Therefore, f must be a constant function.

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