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Math Help - continuity of multivariable functions

  1. #1
    Member Greengoblin's Avatar
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    continuity of multivariable functions

    For proving continuity of single variable functions of the form f:\mathbb{R}\to\mathbb{R} there is the epsilon-delta method, but for multivariable functions of the form f:\mathbb{R}^n\to\mathbb{R} where n\ge 2, we need to show that \lim_{(x_1,...,x_n)\to (a,...,n)}f(x_1,x_2,...,x_n)=f(a,b,...,n) from all possible directions including curves in \mathbb{R}^n that pass through (a,b,...,n) so how do we go about proving continuity?
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    Quote Originally Posted by Greengoblin View Post
    for multivariable functions of the form f:\mathbb{R}^n\to\mathbb{R} where n\ge 2, we need to show that \lim_{(x_1,...,x_n)\to (a,...,n)}f(x_1,x_2,...,x_n)=f(a,b,...,n) from all possible directions including curves in \mathbb{R}^n that pass through (a,b,...,n) so how do we go about proving continuity?
    In another post you said that you are studying basic point set topology.
    But I will answer this question for metric spaces.
    Basically we replace the open interval \left( {a - \delta ,a + \delta } \right) = \left\{ {x:\left| {a - x} \right| < \delta } \right\} with open balls B(p;\delta ) = \left\{ {x:d(x,a) < \delta } \right\}.

    In other words, there is a natural extension of the \varepsilon ;\delta definition.

    As we move into abstract topological spaces, we replace open balls with open sets.

    I hope this very simplified outline gives you some guidance.
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  3. #3
    Member Greengoblin's Avatar
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    Quote Originally Posted by Plato View Post
    In another post you said that you are studying basic point set topology.
    But I will answer this question for metric spaces.
    Basically we replace the open interval \left( {a - \delta ,a + \delta } \right) = \left\{ {x:\left| {a - x} \right| < \delta } \right\} with open balls B(p;\delta ) = \left\{ {x:d(x,a) < \delta } \right\}.

    In other words, there is a natural extension of the \varepsilon ;\delta definition.

    As we move into abstract topological spaces, we replace open balls with open sets.

    I hope this very simplified outline gives you some guidance.
    Ah thanks, I have seen that in the topology notes I'm reading. I've only just started learning point-set topology and so far in my notes I've only read through the preliminaries. I actually didn't know it dealt with continuity of mulitvariable functions, but that gives me more motivation to learn it.
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