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Math Help - Real Analysis: Uniform continuity proof

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    Real Analysis: Uniform continuity proof

    I need some help/advice doing the following proof. Also, I'm in a beginning real analysis class and the section we're covering is on uniform continuity, so keep that in mind I guess.

    Suppose f is uniformly continuous on each of the sets X_1, X_2, ..., X_n and let X = \bigcup_{i=1}^{n} X_i. Show that f need not be continuous on X. Show that, even if f is continuous on X, f need not be uniformly continuous on X.

    I think it should suffice to show the result for X = X_1 \cup X_2, since you could do a simple induction proof to show the result for up to X_n. Other than that though, I'm confused as to where I should even begin. Any help is greatly appreciated.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tonyc4l View Post
    I need some help/advice doing the following proof. Also, I'm in a beginning real analysis class and the section we're covering is on uniform continuity, so keep that in mind I guess.

    Suppose f is uniformly continuous on each of the sets X_1, X_2, ..., X_n and let X = \bigcup_{i=1}^{n} X_i. Show that f need not be continuous on X. Show that, even if f is continuous on X, f need not be uniformly continuous on X.

    I think it should suffice to show the result for X = X_1 \cup X_2, since you could do a simple induction proof to show the result for up to X_n. Other than that though, I'm confused as to where I should even begin. Any help is greatly appreciated.
    What have you tried? You need two things that don't "react" well with one another.
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    Quote Originally Posted by Drexel28 View Post
    What have you tried? You need two things that don't "react" well with one another.
    Well, I saw in a subsequent problem that given the same setup, if you add the condition that each X_i is compact, then you get that f must be uniformly continuous on X. So I'm trying to think of what would happen in this problem if one or both of X_1 or X_2 was not closed or unbounded. I also tried thinking of a counterexample, but I wasn't sure of how general I should be. Could a counterexample just be a specific f, X_1, and X_2?
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