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Thread: differentiability and continuity

  1. #1
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    differentiability and continuity

    (a) Suppose that $\displaystyle f: \Re\rightarrow\Re$is differentiable and, that $\displaystyle f(x)/x\rightarrow 0$ as $\displaystyle x\rightarrow\infty$. Prove that, if $\displaystyle f'(x)\rightarrow k$ as $\displaystyle x\rightarrow\infty$, then $\displaystyle k=0$.
    (b) Give an example of a function $\displaystyle g:\Re\rightarrow\Re$ such that $\displaystyle g(x)\rughtarrow 0$ as $\displaystyle x\rightarrow\infty$, but $\displaystyle g'(x)$ does not tend to a limit as $\displaystyle x\rightarrow\infty$



    (a) I know f(x) must increase slower that x. So for some $\displaystyle N\in\Re$, $\displaystyle x_1,x_2\in\Re$, $\displaystyle |f(x_1)-f(x_2)|<|x_1-x_2|$. Now the trouble of showing that this inequality increases until $\displaystyle \frac{|f(x_1)-f(x_2)|}{x_1-x_2|}<\epsilon$. Dunno what to do!
    (b) I tried all the functions I could think of. I figured it should be oscillating since we dont want the derivative to converge... But it doesnt seem to make since since a converged function/series isnt increasing or decreasing anymore. I thought of something like $\displaystyle (-1)^x/x$ but the derivative is complex. Otherwise I was trying things like sin(1/x) but couldnt find anything that worked.
    Last edited by DontKnoMaff; Nov 18th 2010 at 09:37 AM.
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  2. #2
    MHF Contributor chisigma's Avatar
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    a) it is not limitative to suppose that f(*) is differentiable in 0. In...

    http://www.mathhelpforum.com/math-he...tml#post582679

    ... it has been extablished that if f(*) is differentiable in 0 is...

    $\displaystyle \displaystyle f(x)= f(0) + k\ x + x\ \varepsilon (x) , \lim_{x \rightarrow 0} \varepsilon (x) =0 $ (1)

    From (1) we derive...

    $\displaystyle \displaystyle f^{'}(x)= k + \varepsilon (x) + x\ \varepsilon^{'} (x)$ (2)

    ... and...

    $\displaystyle \displaystyle \frac{f(x)}{x}= k + \varepsilon (x)$ (3)

    ... so that if $\displaystyle \displaystyle \lim_{x \rightarrow \infty} \frac{f(x)}{x} = k $ for (3) is $\displaystyle \displaystyle \lim_{x \rightarrow \infty} \varepsilon(x)=0 $ , if $\displaystyle \displaystyle \lim_{x \rightarrow \infty} f^{'} (x)= k$ for (2) is $\displaystyle \displaystyle \lim_{x \rightarrow \infty} x\ \varepsilon^{'} (x)=0$ and [finally...] if $\displaystyle \displaystyle \lim_{x \rightarrow \infty} \frac{f(x)}{x} = 0$ that means that is $\displaystyle k=0$...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    Last edited by chisigma; Nov 19th 2010 at 03:53 AM.
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  3. #3
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    any help on part b) anybody?
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  4. #4
    MHF Contributor chisigma's Avatar
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    If the Laplace Tranform $\displaystyle \mathcal {L} \{\varphi(t)\}= f(s)$ exists , then the 'Final value theorem' extablishes that...

    $\displaystyle \displaystyle \lim_{t \rightarrow \infty} \varphi(t) = \lim_{s \rightarrow 0} s\ f(s)$ (1)

    Now is...

    $\displaystyle \displaystyle \mathcal{L} \{\varphi^{'} (t)\} = s\ f(s) - \varphi(0)$ (2)

    ... and taking into account (1) and the fact that is $\displaystyle \displaystyle \lim_{t \rightarrow \infty} \varphi(t)=0$ is...

    $\displaystyle \displaystyle \lim_{t \rightarrow \infty} \varphi^{'}(t) = \lim_{s \rightarrow 0} \{ s^{2}\ f(s) - s\ \varphi(0) \} =0$ (3)

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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  5. #5
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    Okay, $\displaystyle f(x)=\frac{\sin(x^3)}{x}$ works.
    Last edited by Jose27; Nov 22nd 2010 at 06:48 PM.
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  6. #6
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by chisigma View Post
    If the Laplace Tranform $\displaystyle \mathcal {L} \{\varphi(t)\}= f(s)$ exists , then the 'Final value theorem' extablishes that...

    $\displaystyle \displaystyle \lim_{t \rightarrow \infty} \varphi(t) = \lim_{s \rightarrow 0} s\ f(s)$ (1)

    Now is...

    $\displaystyle \displaystyle \mathcal{L} \{\varphi^{'} (t)\} = s\ f(s) - \varphi(0)$ (2)

    ... and taking into account (1) and the fact that is $\displaystyle \displaystyle \lim_{t \rightarrow \infty} \varphi(t)=0$ is...

    $\displaystyle \displaystyle \lim_{t \rightarrow \infty} \varphi^{'}(t) = \lim_{s \rightarrow 0} \{ s^{2}\ f(s) - s\ \varphi(0) \} =0$ (3)

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    In Gluskin’s paper ‘Let us teach this generalization of the final-value theorem’ [http://iopscience.iop.org/0143-0807/24/6/005/] there are some important details that usually You don’t find in ordinary complex analysis books. Among others in cap. 4 is written…

    If $\displaystyle \displaystyle \lim_{t \rightarrow \infty} f(t)$ and the Laplace Transform $\displaystyle F(s) = \mathcal{L} \{f(t)\}$ exist, then we have the ‘final value theorem…

    $\displaystyle \displaystyle \lim_{t \rightarrow \infty} f(t) = \lim_{s \rightarrow 0} s\ F(s)$ (4)

    For this equality to be true we assume that f(*) has derivative bounded in $\displaystyle (0,\infty)$ and absolutely integrable in $\displaystyle [0,\infty)$.

    For these reasons what i wrote is valid only under the conditions reported by Gluskin...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    Last edited by chisigma; Nov 23rd 2010 at 11:16 AM. Reason: The post has been fully revised...
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