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Math Help - real analysis hw question

  1. #1
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    Exclamation real analysis hw question

    i need to prove: if f is continuous on EcR and if f(x) is not equal to 0 for some point x in E then there is a nbd Q of x such that f(y) is not equal to 0 for all y in the intersection of Q and E

    any help? thanks so much
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  2. #2
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    Unless there is something that I am missing this can be derived directly from the definition of continuity as follows.

    Since f(x_0)\neq 0 (I will assume that f(x_0)>0, but the other case is similar) let f(x_0)=\alpha. Now choose \epsilon=\frac{\alpha}{2}, and the appropriate \delta. Then for |x-x_0|<\delta you have that 0<f(x_0)-\epsilon<f(x)<\epsilon+f(x_0).
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