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Math Help - function and continuity

  1. #1
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    function and continuity

    When Im trying to do this question , I got stuck

    Prove that f:A-->R^m is continuous ata if and only if each f^i is continuous at a for i=1,2,...,m

    This is my attempt:
    We have to prove 2 sides(=> and <=)

    (=>) Let a belongs to A then f(a) is in R^m
    Since f is continuous at a, we have:
    Given e>0, I can find d>0 such that 0<|x-a|< d implies |f(x)-f(a)|<e
    But each f^i is a scalar field component of f(x) = {(f^1(x),...,f^m(x)}

    My question is can I put f(x) = {(f^1(x),...,f^m(x)} into |f(x)-f(a)|to yield the results?
    Is there any better ideas to do this question

    Thank you so much
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  2. #2
    MHF Contributor

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    Quote Originally Posted by knguyen2005 View Post
    When Im trying to do this question , I got stuck

    Prove that f:A-->R^m is continuous ata if and only if each f^i is continuous at a for i=1,2,...,m

    This is my attempt:
    We have to prove 2 sides(=> and <=)

    (=>) Let a belongs to A then f(a) is in R^m
    Since f is continuous at a, we have:
    Given e>0, I can find d>0 such that 0<|x-a|< d implies |f(x)-f(a)|<e
    But each f^i is a scalar field component of f(x) = {(f^1(x),...,f^m(x)}

    My question is can I put f(x) = {(f^1(x),...,f^m(x)} into |f(x)-f(a)|to yield the results?
    Is there any better ideas to do this question

    Thank you so much
    Yes, that is what you have to do!
    In R^n. |f(x)- f(a)|=\sqrt{(f^1(x)- f^1(a))^2+ ... (f^m(x)- f^m(a))^2}. You should be able to prove that \sqrt{a^2+ b^2+ ...+ z^2}\le |a|+ |b|+ ...+ |c|, perhaps by induction.
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