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Math Help - Intermediate Value Theorem

  1. #1
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    Intermediate Value Theorem

    Hi,

    I would like to know how to proof the following:
    Let f : (a,b) --> Z be continuous on (a,b) where Z is the integers.
    Show that f must be a constant function.

    I know we have to suppose f is nonconstant but next, how do we use the Intermediate Value Theorem to show the complete proof?
    May someone please help.
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    Quote Originally Posted by zxcv View Post
    I would like to know how to proof the following:
    Let f : (a,b) --> Z be continuous on (a,b) where Z is the integers.
    Show that f must be a constant function.
    We use proof by contradiction. Suppose that f\colon(a,\,b)\to\mathbb{Z} is a nonconstant function that is continuous on its domain. Then there must exist c and d in (a,\,b) such that f(c)\neq f(d) (for, otherwise, f would be constant). Choose any noninteger k between f(c) and f(d). Then, since f is a function of a real variable that is continuous on the interval [c,\,d], we may apply the intermediate value theorem to conclude that there is some e\in[c,\,d] such that f(e)=k. But this would mean that the noninteger k is in the codomain of f, which is a contradiction. \square
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  3. #3
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    Hi,

    I was just wondering if we can choose a non integer k between f(c) and f(d) since f is (a,b)-->Z and Z is the integers.
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    Quote Originally Posted by zxcv View Post
    Hi,

    I was just wondering if we can choose a non integer k between f(c) and f(d) since f is (a,b)-->Z and Z is the integers.
    I am not sure I understand what you are asking. f(c) and f(d) must be distinct integers, and there are infinitely many real numbers between any two given integers. Pick one of them, and the intermediate value theorem says that f must take on that value somewhere (which is impossible, since f only gives integral values).
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