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Math Help - continuity

  1. #1
    mms
    mms is offline
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    continuity

    Let f:R->R, suppose there is \L\geqslant 0\ such that for all
    \<br />
x,y \in \mathbb{R}<br />
\<br />
\<br />
\left| {f(x) - f(y)} \right| \leqslant L\left| {x - y} \right|<br />
\<br />

    show that f is continuos.
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  2. #2
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    Quote Originally Posted by mms View Post
    Let f:R->R, suppose there is \L\geqslant 0\ such that for all
    \<br />
x,y \in \mathbb{R}<br />
\<br />
\<br />
\left| {f(x) - f(y)} \right| \leqslant L\left| {x - y} \right|<br />
\<br />

    show that f is continuos.
    Have you made any attempt? Have you seen this condition before? It's known as the Lipschitz condition. It comes up a lot in analysis and topology.
    It is powerful as it specifies a condition on a function which is stronger than being continuous but not as strong as being differentiable (comes in handy for Picard's Theorem and the study of weak solutions of PDEs)
    I recommend googling it and then attempting the question yourself.

    If you get stuck here is a pointer in the right direction:

    Spoiler:
    Choose  \delta = \frac{\epsilon}{L + 1}


    Hope this helps.

    pomp.
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  3. #3
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    Spoiler:

    If we have 0<L<1 then it makes sense to pick what you suggested, since it's L>0 ('cause L=0 makes no sense), and then under this we pick \delta=\frac\epsilon{L}.
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  4. #4
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    Spoiler:
    Suppose, doesn't really matter though. I chose to have L+1 so that the inequality is strict.
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