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Math Help - Continuous function

  1. #1
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    Continuous function

    Suppose fi is continuous from a topological space X to the real numbers,how to prove that g=sup{fi:i=1...n} is also continuous?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by weidingchang View Post
    Suppose fi is continuous from a topological space X to the real numbers,how to prove that g=sup{fi:i=1...n} is also continuous?
    It is quite clear to show that a function is continuous we must only show that it's continuous with respect to some open subbase for the codomain. So, in particular since if K=\left\{(a,\infty):a\in\mathbb{R}\right\} and L=\left\{(-\infty,b):b\in\mathbb{R}\right\} then K\cup L forms an open subbase for \mathbb{R} we must only check the cases for elements of this set.

    If E\in K then E=\left(a,\infty\right) and so g^{-1}\left(E\right)=\left\{x\in X:\max_{1\leqslant j\leqslant n}f_j(x)>a\right\}=\bigcup_{j=1}^{n}\left\{x\in X:f_j(x)>a\right\} and since this is a union of open sets (since each f_j is continuous) it follows that g^{-1}\left(E\right) is continuous. The case for E\in L is analogous.
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    Another question

    How to prove that fg is continuous if both f and g are continuous from a topological space X to the real numbers?Thanks a lot...
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by weidingchang View Post
    How to prove that fg is continuous if both f and g are continuous from a topological space X to the real numbers?Thanks a lot...
    Hint hint:

    Spoiler:


    Try it yourself and let us know what you are having trouble with.

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  5. #5
    Member mabruka's Avatar
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    Quote Originally Posted by weidingchang View Post
    How to prove that fg is continuous if both f and g are continuous from a topological space X to the real numbers?Thanks a lot...

    Try writing down the relation between the inverse images of f,g and fg.
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