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Math Help - a problem on limit

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    a problem on limit

    Let f be a real function continuous and defined in the set D of nonnegative real numbers, such that for each x in D the limit of f(nx) for n tending to infinity is zero (n a positive integer). Prove that the limit of f(x) for x tending to infinity is zero.

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    Quote Originally Posted by paulo1941 View Post
    Let f be a real function continuous and defined in the set D of nonnegative real numbers, such that for each x in D the limit of f(nx) for n tending to infinity is zero (n a positive integer). Prove that the limit of f(x) for x tending to infinity is zero.
    There is a very detailed discussion of this problem here. (It is the second of the two problems discussed there. It is not at all easy. Tim Gowers describes it as "a hard problem for those who have done a first course in analysis (one that perhaps one or two people per year are capable of solving) or a hardish exercise in applying the Baire category theorem.")
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