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Math Help - Unif. Continuity Again

  1. #1
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    Unif. Continuity Again

    If f,g are unif. cont. on an interval [a,b] and g(x) \not= 0 for x \in [a,b], then f/g is unif. cont. on [a,b]

    Being able to get started correctly would be helpful. Thanks.
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  2. #2
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    Quote Originally Posted by Math2010 View Post
    If f,g are unif. cont. on an interval [a,b] and g(x) \not= 0 for x \in [a,b], then f/g is unif. cont. on [a,b]
    The interval [a,b] is closed so any continious function is uniformally continuous.
    First prove that \frac{1}{g} is continuous.
    Without loss of generally, we can assume that g is positive on [a,b].
    Therefore,  \left( {\exists c \in [a,b]} \right)\left( {\forall x \in [a,b]} \right)\left[ {0 < g(c) \leqslant g(x)} \right], the low point theorem.
    Now we have a bound on \frac{1}{g}.
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