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Thread: Topological characterization of Continuity

  1. #1
    Senior Member Danneedshelp's Avatar
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    Topological characterization of Continuity

    "$\displaystyle g$ is continuous if and only if $\displaystyle g^{-1}(A)$ is open whenever $\displaystyle A\in{\mathbb{R}}$ is an open set."

    When is it best to use this definition and what would the steps be to use it?

    I was thinking the implication "if $\displaystyle g^{-1}(A)$ is open whenever $\displaystyle A\in{\mathbb{R}}$ is an open set, then $\displaystyle g$ is continuous," might be helpful for counter examples.

    For example,

    suppose $\displaystyle f:{\mathbb{R}}\rightarrow\\A\in{\mathbb{R}}$ and $\displaystyle f(x)=[[x]]$ (floor function).

    Do I have $\displaystyle f^{-1}(V_{\epsilon}(f(c)))$ produceds an open set (where $\displaystyle V_{\epsilon}(f(c))$ is a $\displaystyle \epsilon$-neighborhood centered at some limit point c of the domain of $\displaystyle f$)?

    Could I use it to show $\displaystyle f(x)=1/x$ is not coninuous at zero?

    I can't find many examples of this characterization of countinuity, so I am finding it hard to work with. I would like to know how to use it for practical continuity problems, as in, showing a particular function is continuous on some interval.

    Thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Danneedshelp View Post
    Could I use it to show $\displaystyle f(x)=1/x$ is not coninuous at zero?
    That is because if $\displaystyle \phi:X\mapsto Y$ is to be continuos as $\displaystyle x=x_0$ then $\displaystyle x_0\in X$. So.....
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  3. #3
    MHF Contributor

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    Notice an important distinction between this definition of "continuous" and the one from Calculus I: in Calculus I we define "continuous at a point" and then say that a function is continuous "on a set" if and only if it is continous at each point of that set. Here, we are defining "continuous" on the entire space. There is no way to talk about a function that is "continous" at some points in the space and not at others.
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