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Math Help - Closed subset

  1. #1
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    Closed subset

    f is a continuous function from [0,1] to [0,1]
    Why is A=\{x\in[0,1]:f(x)\geq x\} closed?
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  2. #2
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    Let the sequence x_n converge to x with x_n\in A for all n. Then f(x) = f(\lim x_n) = \lim f(x_n)\geq \lim x_n = x. So x\in A and A is closed.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bram kierkels View Post
    f is a continuous function from [0,1] to [0,1]
    Why is A=\{x\in[0,1]:f(x)\geq x\} closed?
    Or, note that A=\left\{x\in[,1]:f(x)-x\geqslant 0\right\} and thus if g(x)=f(x)-x then g is (trivially) continuous and A=g^{-1}\left([0,\infty)\right). In fact, although harder to prove it's true that this is the case in any order topology.
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