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Math Help - more continuity of functions

  1. #1
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    more continuity of functions

    1) Let f: D-->R and define |f|: D-->R by |f|(x)=|f(x)|. Suppose that f is continuous at c elements of D. Prove that |f| is continuous at c.

    2) If |f| is continuous at c, does it follow that f is continuous at c? Justify your answer.

    ----I don't know where to start for 1) but for 2), I would think that f is continuous at c since it is part of |f|. Is this proper logic?

    Thanks.
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  2. #2
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    if f is continuous at c then \forall \varepsilon >0 \exists \delta>0 with \vert x - c \vert < \delta \Rightarrow \vert f(x) - f(c) \vert <\varepsilon.
    Using the same \delta, we have
    \vert \vert f(x)\vert - \vert f(c)\vert \vert\leq\vert f(x)-f(c)\vert <\varepsilon, so \vert f\vert is continuous at c.

    For 2) take a function which is -1 on \mathbb{Q} and 1 on \mathbb{R}-\mathbb{Q}. Then \vert f \vert is 1 and continous, but f certainly isn't.
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