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Thread: closed sets and continuity

  1. #1
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    closed sets and continuity

    Let $\displaystyle f:R \rightarrow R$ be any 1-variable function. Define $\displaystyle A=\{(x,y)\in R^2:y\leq f(x)\}$ and $\displaystyle B=\{(x,y) \in R^2 : y\geq f(x)\}$.
    Prove that $\displaystyle f$ is continuous $\displaystyle iff$ $\displaystyle A$ and $\displaystyle B$ are closed in $\displaystyle R^2$.
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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 $\displaystyle (x_n,y_n)$ be a sequence of points in $\displaystyle \mathbb{R}^2$. Then $\displaystyle (x_n,y_n)\rightarrow (x,y)$ if and only if $\displaystyle x_n\rightarrow x$ and $\displaystyle y_n\rightarrow y$.

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