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Math Help - right limit of a continuous bounded function

  1. #1
    Senior Member abhishekkgp's Avatar
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    right limit of a continuous bounded function

    i was thinking whether the following is true or false:

    Let f: (a,b)  \rightarrow \mathbb{R} be a continuous and bounded function. then, f has a right limit at a.(true or false?).

    i thought that this could be proved to be true if i were able to show that the function is monotonic in some deleted right neighborhood of a but then sin(1/x) served as a counter example as the claim would be violated at x=0.
    intuitively it seems that the right limit at a should exist but i am not able to prove it. please help.
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  2. #2
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    Quote Originally Posted by abhishekkgp View Post
    i was thinking whether the following is true or false: Let f: (a,b)  \rightarrow \mathbb{R} be a continuous and bounded function. then, f has a right limit at a.(true or false?).

    i thought that this could be proved to be true if i were able to show that the function is monotonic in some deleted right neighborhood of a but then sin(1/x) served as a counter example as the claim would be violated at x=0.
    intuitively it seems that the right limit at a should exist but i am not able to prove it. please help.
    Haven't you answered your own question?
    \lim _{x \to 0^ +  } \sin \left( {\frac{1}{x}} \right) = ?
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  3. #3
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by Plato View Post
    Haven't you answered your own question?
    \lim _{x \to 0^ +  } \sin \left( {\frac{1}{x}} \right) = ?
    oh yes.. my bad.. thanks!!
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