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Math Help - how can I prove if this function is continuous or not!

  1. #1
    Junior Member
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    how can I prove if this function is continuous or not!

    <br />
   f(x) = \left\{<br />
     \begin{array}{lr}<br />
         x^2 sin\frac{1}{x} & : x \neq 0\\<br />
       0 & : x = 0<br />
     \end{array}<br />
   \right.<br />
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  2. #2
    Super Member redsoxfan325's Avatar
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    Swampscott, MA
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    Quote Originally Posted by binkypoo View Post
    <br />
   f(x) = \left\{<br />
     \begin{array}{lr}<br />
         x^2 sin\frac{1}{x} & : x \neq 0\\<br />
       0 & : x = 0<br />
     \end{array}<br />
   \right.<br />
    Take the limit: -\lim_{x\to0}x^2\leq\lim_{x\to0}x^2\sin(1/x)\leq\lim_{x\to0}x^2

    So \lim_{x\to0}x^2\sin(1/x)=0=f(0), and it's continuous at x=0.

    It's also differentiable at x=0.
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  3. #3
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    doesnt this show that f is continuous at x = 0? What about the rest of the domain?
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  4. #4
    Super Member redsoxfan325's Avatar
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    It's continuous everywhere else because x^2\sin(x) and \frac{1}{x} are. The composition of continuous functions is continuous.
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  5. #5
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    I guess my problem is I dont know how to show that x^2sin(x) and 1/x are continuous EVERYWHERE. Could you explain that please? thanks
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  6. #6
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by binkypoo View Post
    I guess my problem is I dont know how to show that x^2sin(x) and 1/x are continuous EVERYWHERE. Could you explain that please? thanks
    I don't think your professor expects you to do all that, but if he does, you need to crank out the \epsilon- \delta proofs for each one.

    For x^2 let \delta=\min\left\{1,\frac{1}{1+2|x_0|}\right\}, because |x-x_0|<1\implies |x+x_0|<1+2|x_0|\implies |x-x_0||x+x_0|<\delta(1+2|x_0|)

    For \sin x, you can probably Google a proof. The idea is to let \delta=\epsilon and then use a difference-to-product identity to convert |\sin x-\sin x_0| into something more manageable.

    \frac{1}{x} is kind of a pain in the ass, and I've screwed up continuity proofs of this function before. There are probably other threads on MHF with proofs of this.
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