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Math Help - Continuity definiton problem

  1. #1
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    Continuity definiton problem

    Show that f: A \rightarrow R^{m} is continuous at x_0 iff for every  \epsilon > 0 , there exist  \delta > 0 such that  || x-x_0 || \leq \delta , we have  ||f(x)-f(x_0)|| \leq \epsilon

    So I can replace the definition < with  \leq ?
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Show that f: A \rightarrow R^{m} is continuous at x_0 iff for every  \epsilon > 0 , there exist  \delta > 0 such that  || x-x_0 || \leq \delta , we have  ||f(x)-f(x_0)|| \leq \epsilon
    So I can replace the definition < with  \leq ?
    I am not sure that I follow your question.
    Do note that the tradition definition is for <.
    However, it is also \left( {\forall \varepsilon  > 0} \right)\left[ {\frac{\varepsilon }<br />
{2} < \varepsilon } \right], that is for all.
    So \left\| {f(x) - f(x_0 )} \right\| \leqslant \frac{\varepsilon }<br />
{2} \Rightarrow \quad \left\| {f(x) - f(x_0 )} \right\| < \varepsilon
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  3. #3
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    I asked the professor today and he said this problem is suppose to show that the two definitions for continuity are equiv.

    Well, from the last post I understand how to go from  \leq to <

    How to do this the other way around is killing me...

    Any hints? Thanks.
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