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Math Help - Continuity Problem

  1. #1
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    Continuity Problem

    1) If g(x) is continuous at c and g(c)>0, show that there is an open interval centered at c on which g(x) is always positive.

    It makes sense when I read it, I'm confused how I go about proving it.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Hint : let \epsilon = g(c)/2 in the definition of continuity at c.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Janu42 View Post
    1) If g(x) is continuous at c and g(c)>0, show that there is an open interval centered at c on which g(x) is always positive.

    It makes sense when I read it, I'm confused how I go about proving it.
    The definition of continuity. g(c)\in(0,\infty) and since this is an open set there exists some B_{\delta}(g(c))\subseteq (0,\infty) and by continuity there exists some B_{\delta}(c) such that g(B_{\delta}(c))\subseteq B_{\varepsilon}(g(c))\subseteq(0,\infty)
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