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Math Help - closed sets and continuity

  1. #1
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    closed sets and continuity

    Let f:R \rightarrow R be any 1-variable function. Define A=\{(x,y)\in R^2:y\leq f(x)\} and B=\{(x,y) \in R^2 : y\geq f(x)\}.
    Prove that f is continuous iff A and B are closed in R^2.
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  2. #2
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    Use the sequence definitions! Keep in mind a sequence of coordinate pairs converges if and only if the individual coordinates converge, and relate that to the convergence-preserving properties of continuous functions.
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  3. #3
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    closed set

    where do you want a sequence to converge?
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  4. #4
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    Quote Originally Posted by Kat-M View Post
    where do you want a sequence to converge?
    Formally, here are the two qualities I was talking about:

    Let (x_n,y_n) be a sequence of points in \mathbb{R}^2. Then (x_n,y_n)\rightarrow (x,y) if and only if x_n\rightarrow x and y_n\rightarrow y.

    Let f:\mathbb{R}\rightarrow\mathbb{R}. Then f is continuous at x if and only if for every sequence x_n\rightarrow x, f(x_n)\rightarrow f(x).
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