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Math Help - f(x)=x*Sin(1/x) uniformly continuous

  1. #1
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    f(x)=x*Sin(1/x) uniformly continuous

    Hello,


    I want to show that
    f(x)=x \cdot \sin( \frac{1}{x})
    in the interval (0, infinity)
    is uniformly continuous using the following definition:

     \text{Given }f \colon I \subset \mathbf{R} \longrightarrow \mathbf{R}. f \\<br />
\text{ is uniformly continuous on I if }\forall \varepsilon >0, \exists \delta >0 \\<br />
\text{ such that }\forall x,y \in I, | x - y| < \delta, |f(x)-f(y)|< \varepsilon.

    My problem is that I want to find something like:
    |f(x)-f(y)|\leq \underbrace{|x-y|}_{ \delta }\dots \leq \varepsilon
    but I've been thinking for a while and I cant find anything.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by sunmalus View Post
    Hello,


    I want to show that
    f(x)=x \cdot \sin( \frac{1}{x})
    in the interval (0, infinity)
    is uniformly continuous using the following definition:

     \text{Given }f \colon I \subset \mathbf{R} \longrightarrow \mathbf{R}. f \\<br />
\text{ is uniformly continuous on I if }\forall \varepsilon >0, \exists \delta >0 \\<br />
\text{ such that }\forall x,y \in I, | x - y| < \delta, |f(x)-f(y)|< \varepsilon.

    My problem is that I want to find something like:
    |f(x)-f(y)|\leq \underbrace{|x-y|}_{ \delta }\dots \leq \varepsilon
    but I've been thinking for a while and I cant find anything.
    Two things to notice. f may be continuous extended to \hat{f}:[0,\infty)\to\mathbb{R} in the usual (and unique) way. Thus, for any M>0 \hat{f} is continuous on [0,M] and thus by the Heine-Cantor Theorem we see that \hat{f} is unif. cont. on [0,M]. Thus, any restriction of \hat{f} on [0,M] is unif. cont., in particular f is unif. cont. on (0,M]. But, \displaystyle \lim_{x\to\infty}f(x)=1... so try working with that.

    If you can use something besides the strict definition and consequences I would note that f is unif. cont. on (0,1] as proven above and f is Lipschitz on (1,\infty) since it has bounded derivative. From there you can conclude.
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    This is one of my favorite examples in infi 1.

    Let \epsilon >0 and \delta =\frac{\epsilon }{2}. Let x_0,x_1 \in \mathbb{R}. Now we suppose that \left | x_0-x_1 \right | <\delta .

    Let we analyze \left | f(x_0)-f(x_1) \right | .

     \left | f(x_0)-f(x_1) \right |=\left | x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_1}) \right |

    \leq \left |x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_0})+x_0sin(\frac{1}{x_1})-x_1sin(\frac{1}{x_1})  \right |

    =\left | x_0-x_1 \right |\left | sin(\frac{1}{x_0})+sin(\frac{1}{x_1}) \right |<br />
    \leq \left | x_0-x_1|\left \{  \right. \right  \left | sin(\frac{1}{x_0}) \right |+\left | sin(\frac{1}{x_1}) \right | \left.  \right \}\right.\left.  <br />
    \leq 2\left | x_0-x_1|\left < \epsilon
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    This is one of my favorite examples in infi 1.

    Let \epsilon >0 and \delta =\frac{\epsilon }{2}. Let x_0,x_1 \in \mathbb{R}. Now we suppose that \left | x_0-x_1 \right | <\delta .

    Let we analyze \left | f(x_0)-f(x_1) \right | .

     \left | f(x_0)-f(x_1) \right |=\left | x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_1}) \right |

    \leq \left |x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_0})+x_0sin(\frac{1}{x_1})-x_1sin(\frac{1}{x_1})  \right |

    =\left | x_0-x_1 \right |\left | sin(\frac{1}{x_0})+sin(\frac{1}{x_1}) \right |<br />
    \leq \left | x_0-x_1|\left \{  \right. \right  \left | sin(\frac{1}{x_0}) \right |+\left | sin(\frac{1}{x_1}) \right | \left.  \right \}\right.\left.  <br />
    \leq 2\left | x_0-x_1|\left < \epsilon
    Nice! I didn't even try the usual methods assuming it would get too messy. I forgot about the bounded continuous function times a uniformly continuous function thing.
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  5. #5
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    Thanks!
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  6. #6
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    Thanks!
    I forgot to mention that \sin(a) + \sin(b)= 2\sin(\frac{a+b}{2}) \cos(\frac{a-b}{2}) was given as "indication", so I was trying to find a way to use it and that's what confused me the most. (I still don't see how to use this tho).

    Well it might be useful if you're using series to prove this but with this definition :S .
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  7. #7
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    Quote Originally Posted by Also sprach Zarathustra View Post
    This is one of my favorite examples in infi 1.

    Let \epsilon >0 and \delta =\frac{\epsilon }{2}. Let x_0,x_1 \in \mathbb{R}. Now we suppose that \left | x_0-x_1 \right | <\delta .

    Let we analyze \left | f(x_0)-f(x_1) \right | .

     \left | f(x_0)-f(x_1) \right |=\left | x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_1}) \right |

    \leq \left |x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_0})+x_0sin(\frac{1}{x_1})-x_1sin(\frac{1}{x_1})  \right |

    =\left | x_0-x_1 \right |\left | sin(\frac{1}{x_0})+sin(\frac{1}{x_1}) \right |<br />
    \leq \left | x_0-x_1|\left \{  \right. \right  \left | sin(\frac{1}{x_0}) \right |+\left | sin(\frac{1}{x_1}) \right | \left.  \right \}\right.\left.  <br />
    \leq 2\left | x_0-x_1|\left < \epsilon
    Dear Also sprach Zarathustra,

    Can you please tell me how you got,

    \left | x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_1}) \right |\leq \left |x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_0})+x_0sin(\frac{1}{x_1})-x_1sin(\frac{1}{x_1}) \right |

    Thank you.
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  8. #8
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by Sudharaka View Post
    Dear Also sprach Zarathustra,

    Can you please tell me how you got,

    \left | x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_1}) \right |\leq \left |x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_0})+x_0sin(\frac{1}{x_1})-x_1sin(\frac{1}{x_1}) \right |

    Thank you.

    You caught me!

    Unfortunately, I can't find my notes where I "solved" it...


    I will try to explain my work anyway:


    \left | x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_1}) \right |= \left |x_0sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_0})+x_1sin(\frac{1}{x_0})-x_1sin(\frac{1}{x_1}) \right |


    Now because, \left | x_0-x_1 \right | <\delta<br />

    We can conclude that: x_1sin(\frac{1}{x_0})\simeq x_0sin(\frac{1}{x_1})

    hmmm...
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