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Thread: Prove This Is Discontinuous At Zero

  1. #1
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    Prove This Is Discontinuous At Zero

    Let E be a normed vector space. Let $\displaystyle A:E\longrightarrow E$ be defined by $\displaystyle A(p(x))=p'(x)$, the derivative of $\displaystyle p(x)$. Show that the $\displaystyle \epsilon, \delta$ definition of a continuous function fails at zero.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathematicalbagpiper View Post
    Let E be a normed vector space. Let $\displaystyle A:E\longrightarrow E$ be defined by $\displaystyle A(p(x))=p'(x)$, the derivative of $\displaystyle p(x)$. Show that the $\displaystyle \epsilon, \delta$ definition of a continuous function fails at zero.
    This makes no sense. I assume that $\displaystyle E$ is some kind of polynomial space. What kind of norm? The sup norm? You need be much more specific.
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    I just looked up the reference in the textbook (he references an earlier remark in the book). Here's a little more relevant information:

    Let $\displaystyle E=\mathbb{R}[x]$.

    So $\displaystyle p(x)\in\mathbb{R}[x]$.

    Further, the norm on this space he defined as $\displaystyle ||p||=max\{|a_0|,......,|a_n|\}$

    I hope that's enough.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathematicalbagpiper View Post
    I just looked up the reference in the textbook (he references an earlier remark in the book). Here's a little more relevant information:

    Let $\displaystyle E=\mathbb{R}[x]$.

    So $\displaystyle p(x)\in\mathbb{R}[x]$.

    Further, the norm on this space he defined as $\displaystyle ||p||=max\{|a_0|,......,|a_n|\}$

    I hope that's enough.
    Ok, so what have you tried? So when you say zero I assume you mean $\displaystyle 0(x)=0+0+\cdots$, right? So, you need to find some $\displaystyle \varepsilon>0$ such that given any $\displaystyle \delta>0$ there is some $\displaystyle p(x)\in\mathbb{R}[x]$ such that $\displaystyle \|p(x)\|<\delta$ but $\displaystyle \|p'(x)\|\geqslant \varepsilon$. So what do you think?
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    Honestly, I haven't a bloody clue, and just out of curiosity, why is it just that $\displaystyle ||p(x)||<\delta$ but $\displaystyle ||p'(x)||\geq\epsilon$ instead of $\displaystyle ||p(x)-p(a)||<\delta$ but $\displaystyle ||p'(x)-p'(a)||\geq\epsilon$?

    Sorry, I know I should know this, but I barely scraped by analysis in undergrad :-(
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  6. #6
    Senior Member roninpro's Avatar
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    You should try writing down the definition of continuity for your map $\displaystyle A$:

    Given $\displaystyle \varepsilon>0$ there is $\displaystyle \delta>0$ so that $\displaystyle ||p(x)-0||<\delta$ implies $\displaystyle ||A(p(x))-A(0)||<\varepsilon$.

    When you clean this up, you get what Drexel suggested.
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  7. #7
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathematicalbagpiper View Post
    Honestly, I haven't a bloody clue, and just out of curiosity, why is it just that $\displaystyle ||p(x)||<\delta$ but $\displaystyle ||p'(x)||\geq\epsilon$ instead of $\displaystyle ||p(x)-p(a)||<\delta$ but $\displaystyle ||p'(x)-p'(a)||\geq\epsilon$?

    Sorry, I know I should know this, but I barely scraped by analysis in undergrad :-(
    Forget $\displaystyle \varepsilon$, we can do this for $\displaystyle 1$. So ,if $\displaystyle A:\mathbb{R}[x]\to\mathbb{R}[x]$ were continuous at $\displaystyle 0(x)$ then we should be able to find some $\displaystyle \delta>0$ such that $\displaystyle \|p(x)-0(x)\|=\|p(x)\|<\delta\implies \|p'(x)-0'(x)\|=\|p'(x)\|<1$, right? But, for example if you claimed to me that you have found a $\displaystyle 2\delta_0$ which works I can show you that you're wrong. To see this we know by the Archimedean principle I can find some $\displaystyle N\in\mathbb{N}$ such that $\displaystyle N\delta_0>1$ and $\displaystyle N\geqslant2$, right? So, consider $\displaystyle p(x)=\delta_0x^N$ then clearly $\displaystyle \|p(x)\|=\max\{\delta_0\}=\delta_0<2\delta_0$ but $\displaystyle p'(x)=N\delta_0x^{N-1}$, and $\displaystyle \|p'(x)\|=\|N\delta_0x^{N-1}\|=N\delta_0>1$ and so the $\displaystyle \delta=2\delta_0$ you claimed work didn't, and so you can't find any $\displaystyle \delta$ which works. So, it isn't continuous. Make sense?
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  8. #8
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    Now that makes sense. I think I have an idea where to go now.
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