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A continuous function problem

  1. Feb 14, 2012 #1
    1. The problem statement, all variables and given/known data
    assume a function F(x)=(a|x|^(a-1))*(sin(1/x))-((|x|^a)/(x^2))*(cos(1/x)) for x not equal to 0
    F(x)=0 for x equal to 0

    for what values of a that this function is continuous on R(real number)


    2. Relevant equations
    the F(x) is the differentiation of |x|^a sin(1/x)

    3. The attempt at a solution
    i don.t know how to consider the value a
     
  2. jcsd
  3. Feb 14, 2012 #2

    SammyS

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    Treat the variable, a, as a constant.

    What must be true in order for F(x) to be continuous at x?
     
  4. Feb 14, 2012 #3
    i am consider that if both the two parts of the function can be differentiable then both of then are continuous,then done. but how i know if a>3 then they are both differentiable at 0, but how about the other points. does this method make sense ?
     
  5. Feb 14, 2012 #4

    SammyS

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    OK: So we have
    [itex]F(x)=\left\{ \begin{array}{cc} 0\,,&\text{ if }x=0 \\ a|x|^{a-1}\sin(1/x)-(|x|^a/x^2)\cos(1/x)\,,&\text{ otherwise} \end{array} \right. [/itex]​
    All the functions of which F(x) is composed are continuous for all x except some are not continuous at x = 0.

    So It appears that you need to see what values of the variable, a, makes F(x) continuous at x=0.

    What's the test to see if F(x) is continuous at x=0 ?
     
  6. Feb 15, 2012 #5
    that means lim x->0 F(x) exists,right? then i can prove it
     
  7. Feb 15, 2012 #6

    SammyS

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    Nothing I wrote shows that F(x) is continuous at x=0 !

    Does [itex]\displaystyle \lim_{x\to0}\,F(x)[/itex] exist?

    If so, is [itex]\displaystyle \lim_{x\to0}\,F(x)=F(0)\,,\ \text{ which is }0\,?[/itex]

    To answer yes to these questions may impose restrictions on the value of the variable, a,
     
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