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Math Help - Functions

  1. #1
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    Functions

    Please help me thanks.

    Let
    f : R - {0} ---> Rbe defined by

    f
    (x) = x^asin(x^-b);

    where
    b > 0. Find conditions on a; b such that
    (a)
    f can be extended to a continuous function on R.
    (b)
    f can be extended to a differentiable function on R.

    (c)
    f can be extended to a twice differentiable function on R.

    R is the set of real numbers in this case.
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  2. #2
    Member HappyJoe's Avatar
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    What needs to be done in part (a) is to check for which values of a and b does the function f have a limit for x tending to 0. If f does have a (finite) limit, then you can extend f to all of R by letting f(0) be that limit.

    As for part b and c, you do something very similar. Well, first check if the function f is differentiable in the first place. You know that if f is to be extended to a differentiable function, then the value of f at 0 must be the limit of f(x) for x-->0. Check which choices of a and b make the extension differentiable at 0. Similarly for the second derivative.
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  3. #3
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    Quote Originally Posted by HappyJoe View Post
    What needs to be done in part (a) is to check for which values of a and b does the function f have a limit for x tending to 0. If f does have a (finite) limit, then you can extend f to all of R by letting f(0) be that limit.

    As for part b and c, you do something very similar. Well, first check if the function f is differentiable in the first place. You know that if f is to be extended to a differentiable function, then the value of f at 0 must be the limit of f(x) for x-->0. Check which choices of a and b make the extension differentiable at 0. Similarly for the second derivative.
    Forgive me for being not smart enough. Could you elaborate more? Like how to check which values of a and b will f have a limit for x tending to 0?
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  4. #4
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    Quote Originally Posted by ineedyourhelp View Post
    Please help me thanks.


    Let f : R - {0} ---> Rbe defined by

    f
    (x) = x^asin(x^-b);
    where


    b > 0. Find conditions on a; b such that
    (a) f can be extended to a continuous function on R.
    (b) f can be extended to a differentiable function on R.
    (c) f can be extended to a twice differentiable function on R.

    R is the set of real numbers in this case.

    Another question from an assignment that counts towards your final grade?

    Thread closed.

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