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Thread: Not Uniform Continuous?

  1. #1
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    Not Uniform Continuous?

    Consider the function $\displaystyle f(x)=xsinx$

    Now, $\displaystyle f'(x)=xcosx+sinx$, which is not bounded, so it is not uniformly continuous.

    So pick $\displaystyle \epsilon = 1 $, let $\displaystyle \delta > 0$, for every $\displaystyle x,y \in \mathbb {R} , |x-y| < \delta $, we need $\displaystyle |xsinx-ysiny| \geq \epsilon $, but how should I go about that? Thanks.
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  2. #2
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    "$\displaystyle f$ is not uniformly continuous" means:
    $\displaystyle \exists\varepsilon_0>0, \forall \delta>0, \exists x,y$ such that $\displaystyle |x-y|<\delta$ and $\displaystyle |f(x)-f(y)|>\varepsilon_0$.

    I let you get convinced yourself that a proof would consist in finding a sequence a couples $\displaystyle (x_n,y_n)_{n\geq 1}$ such that $\displaystyle |x_n-y_n|\to_n 0$ and $\displaystyle |f(x_n)-f(y_n)|>\varepsilon_0$ for some $\displaystyle \varepsilon_0$ (or even $\displaystyle |f(x_n)-f(y_n)|\to_n+\infty$, if possible).

    In order to choose those couples, look at (or just imagine) what the graph looks like: the $\displaystyle x_n,y_n$ must be close but the $\displaystyle f(x_n),f(y_n)$ must be way apart from each other.

    $\displaystyle x_n=n\pi$ is a possible start.
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  3. #3
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    Thanks for the hint. So for the choice that $\displaystyle x_n = n \pi $, we would have $\displaystyle f(x_n) = 0 \ \ \ \forall n $, but any point right afterward would get larger as n increases.

    Pick $\displaystyle y_n = x_n + \frac { \delta}{2} $, then $\displaystyle |x_n - y_n | < \delta $, but $\displaystyle |f(x_n)-f(y_n)| \rightarrow \infty $

    Is this right?
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  4. #4
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    Quote Originally Posted by tttcomrader View Post
    Pick $\displaystyle y_n = x_n + \frac { \delta}{2} $, then $\displaystyle |x_n - y_n | < \delta $, but $\displaystyle |f(x_n)-f(y_n)| \rightarrow \infty $

    Is this right?
    No, because for every $\displaystyle \delta>0$ you must have $\displaystyle |x_n-y_n|<\delta$ for some $\displaystyle n\geq 0$. This is why I said you must have $\displaystyle |x_n-y_n|\to_n 0$.

    You can write $\displaystyle y_n=x_n+u_n=n\pi+u_n$ where $\displaystyle u_n\to_n 0^+$ is to be determined.

    You have $\displaystyle |f(y_n)-f(x_n)|=|f(y_n)|=|(n\pi+u_n)\sin(n\pi+u_n)|=|(n\pi +u_n)(-1)^n\sin(u_n)|$$\displaystyle \sim n\pi\sin(u_n)\sim n\pi u_n$ as $\displaystyle n\to\infty$. So that you must choose $\displaystyle u_n\to_n 0$ such that $\displaystyle \liminf_n n u_n>0$. For instance, $\displaystyle u_n=\frac{1}{n}$ (hence $\displaystyle \varepsilon_0$ must be small enough), or $\displaystyle u_n=\frac{1}{\sqrt{n}}$ (and any $\displaystyle \varepsilon_0$ is adequate).

    You can do a more explicit proof using the convexity inequality $\displaystyle \sin(u_n)\geq\frac{2u_n}{\pi}$ which holds as soon as $\displaystyle 0\leq u_n\leq \frac{\pi}{2}$. This yields to the same possibilities for the choice of $\displaystyle (u_n)_n$.
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  5. #5
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    Thanks for the help. But one thing I'm not clear is, why is it enough to show that $\displaystyle
    |x_n-y_n|\to_n 0
    $ implies $\displaystyle
    |f(x_n)-f(y_n)|>\varepsilon_0
    $?
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  6. #6
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    Explicitating my first post as much as I can:

    You know that "$\displaystyle f$ is not uniformly continuous" means:
    $\displaystyle \exists\varepsilon_0>0, \forall \delta>0, \exists x,y$ such that $\displaystyle |x-y|<\delta$ and $\displaystyle |f(x)-f(y)|>\varepsilon_0$.
    You obtain this by taking the logical contrary of the definition of "uniformly continuous".

    Suppose we found a sequence of couples $\displaystyle (x_n,y_n)_{n\geq 1}$ and an $\displaystyle \varepsilon_0>0$ such that $\displaystyle |x_n-y_n|\to_n 0$ and, for all $\displaystyle n\geq 1$, $\displaystyle |f(x_n)-f(y_n)|>\varepsilon_0$.

    Then, for every $\displaystyle \delta>0$, there is an $\displaystyle n$ such that $\displaystyle |x_n-y_n|<\delta$ (because of the convergence to 0). And for this same $\displaystyle n$, we have $\displaystyle |f(x_n)-f(y_n)|>\varepsilon_0$. As a consequence, we have realized the above definition of "not uniformly continuous" with $\displaystyle x_n,y_n$ in place of $\displaystyle x,y$.
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