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Math Help - Continuous Problem

  1. #1
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    Continuous Problem

    Suppose that f and g are defined and finite valued on an open interval I which contains a, that f is cont. at a, and that f(a) =/= 0. Then g is cont. at a if and only if fg is cont. at a.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Math2010 View Post
    Suppose that f and g are defined and finite valued on an open interval I which contains a, that f is cont. at a, and that f(a) =/= 0. Then g is cont. at a if and only if fg is cont. at a.
    I mean, what have you done? Is fg multiplication or composition?
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  3. #3
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    fg is multiplication.

    I know we are given \forall \epsilon > 0, \exists \delta_f > 0, \delta_g > 0 s.t x,a \in A

    |x - a| < \delta_f, |x - a| < \delta_g \forall x,a \in A \Rightarrow |f(x)-f(a)|<\epsilon, |g(x)-g(a)|<\epsilon

    I think I can simplify |fg(x) - fg(a)| pretty easily to get it to equal f(x)|g(x) - g(a)| + g(a)|f(x)-f(a)|

    I understand that |g(x) - g(a)| and |f(x)-f(a)| represent epsilon so we have |fg(x) - fg(a)|\le f(x)\epsilon + g(a)\epsilon but now is where I really do not know what I need to do for f(x) and g(a). Thank you for the help.
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