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Math Help - Continuity and Limits Proofs

  1. #1
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    Continuity and Limits Proofs

    Question: Give an example of a function g such that g is continuous nowhere, but |g| is continuous everywhere?

    Fairly simple set-up. I'm thinking that the best way to attack this would be with a piece-wise function (similar to the classic 1/q if rational, 0 if irrational), but I'm having trouble thinking of a function whose absolute value WOULD be continuous. I've done a couple of these problems with properties before, but I just am really stumped on this one. Any help is appreciated!

    Thanks again,

    Mike
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by mscbuck View Post
    Question: Give an example of a function g such that g is continuous nowhere, but |g| is continuous everywhere?

    Fairly simple set-up. I'm thinking that the best way to attack this would be with a piece-wise function (similar to the classic 1/q if rational, 0 if irrational), but I'm having trouble thinking of a function whose absolute value WOULD be continuous. I've done a couple of these problems with properties before, but I just am really stumped on this one. Any help is appreciated!

    Thanks again,

    Mike
    Hmmm, well, the most notable thing about absolute values is that they make negative numbers positive. So going off your idea, how about g(x) = 1 if x is rational and g(x) = -1 if x is irrational?
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