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Math Help - counterexamples in real analysis

  1. #1
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    counterexamples in real analysis

    Prove or give a counterexample for each statement
    A) If f^2 is continuous on D, then f is continuous on D
    B) If f is continuous on D, then f(D) is a bounded subset
    C) if f and g are not continuous on D, then f + g is not continuous on D
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  2. #2
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    Quote Originally Posted by luckyc1423 View Post
    Prove or give a counterexample for each statement
    A) If f^2 is continuous on D, then f is continuous on D
    Yes.

    If f(x)>=0 is continous.
    Then, sqrt(f(x)) is continous.

    Then,
    sqrt(f^2)=|f| is continous.
    But if |f| is contonous it must be that f was continous.
    B) If f is continuous on D, then f(D) is a bounded subset
    False.
    Consider,
    1/x on (0,1)
    C) if f and g are not continuous on D, then f + g is not continuous on D
    I have an elegant approach to this one.
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