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Math Help - weaker condition for continuity.

  1. #1
    Senior Member abhishekkgp's Avatar
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    weaker condition for continuity.

    Let f:\mathbb{R} \rightarrow \mathbb{R}, \, a\in S \subset \mathbb{R}.Suppose f has the property that:
    x_n \in S, \, x_n \rightarrow a \Rightarrow (f(x_n)) is convergent. Prove that f is continuous at a.

    my approach:
    Let (x_n) and (y_n) be two sequences in S with x_n \rightarrow a, \, y_n \rightarrow a. let f(x_n) \rightarrow L_1 and f(y_n) \rightarrow L_2. I need to prove that L_1=L_2. how do i go about it?
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  2. #2
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    Quote Originally Posted by abhishekkgp View Post
    Let f:\mathbb{R} \rightarrow \mathbb{R}, \, a\in S \subset \mathbb{R}.Suppose f has the property that:
    x_n \in S, \, x_n \rightarrow a \Rightarrow (f(x_n)) is convergent. Prove that f is continuous at a.

    my approach:
    Let (x_n) and (y_n) be two sequences in S with x_n \rightarrow a, \, y_n \rightarrow a. let f(x_n) \rightarrow L_1 and f(y_n) \rightarrow L_2. I need to prove that L_1=L_2. how do i go about it?

    I think this is false. Take for example \displaystyle{f(x):=\left\{\begin{array}{cc}\frac{  \sin x}{x}&\mbox{ , if } x\neq 0\\{}\\ 8&\mbox { , if } x=0\end{array}\right.} , and let

    a\in S:=(-1,1)\subset \mathbb{R} .

    Tonio
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  3. #3
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    Quote Originally Posted by abhishekkgp View Post
    Let f:\mathbb{R} \rightarrow \mathbb{R}, \, a\in S \subset \mathbb{R}.Suppose f has the property that:
    x_n \in S, \, x_n \rightarrow a \Rightarrow (f(x_n)) is convergent. Prove that f is continuous at a.

    my approach:
    Let (x_n) and (y_n) be two sequences in S with x_n \rightarrow a, \, y_n \rightarrow a. let f(x_n) \rightarrow L_1 and f(y_n) \rightarrow L_2. I need to prove that L_1=L_2. how do i go about it?
    You can do this by considering the sequence x_1,y_1,x_2,y_2,x_3,y_3,\ldots.

    Notice that it is not sufficient just to show that L_1=L_2. You need to show that this limit is equal to f(a). You can do that by taking one of your sequences to be the constant sequence x_n=a for all n.
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  4. #4
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by tonio View Post
    I think this is false. Take for example \displaystyle{f(x):=\left\{\begin{array}{cc}\frac{  \sin x}{x}&\mbox{ , if } x\neq 0\\{}\\ 8&\mbox { , if } x=0\end{array}\right.} , and let

    a\in S:=(-1,1)\subset \mathbb{R} .

    Tonio
    a typo error in my question.... i had written f:\mathbb{R} \rightarrow \mathbb{R} while it should be f:S \rightarrow \mathbb{R}.
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