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Math Help - Uniform continuity

  1. #1
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    Uniform continuity

    Before i posted this,i tried reading my notes again and again! so any help....id be thankful!
    here is the problem...

    consider f(x) = lnx, with A element in (0,1)
    confirm the continuity of f in (0,1)
    Let x0 be an element in (0,1)
    then |lnx - lnx0| = | ln x/x0 | = ln(1 - (x0 - x)/x0)
    this is smaller than or equal to 2|(x0 - x)/x0| if |(x0 - x)/x0| < 1/2

    ok so first.. why is |(x0 - x)/x0| < 1/2?? did someone decide to randomnly pick 1/2...? and why is |(x0 - x)/x0| < 1/2... where did the ln go..? wouldnt it make more sense if it was .. ln|(x0 - x)/x0| < 1/2 ?

    Take delta = min (epsilon*(x0)/2 , 1/2*x0)

    how was delta chosen to be these??

    thank you! any help ... would save me a lot of time..effort. im looking through books..online.. believe me im trying but this is logic (and i am poor at it!)

    i havent written up the whole problem,only the stage i am stuck so far!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by matlabnoob View Post
    Before i posted this,i tried reading my notes again and again! so any help....id be thankful!
    here is the problem...

    consider f(x) = lnx, with A element in (0,1)
    confirm the continuity of f in (0,1)
    Let x0 be an element in (0,1)
    then |lnx - lnx0| = | ln x/x0 | = ln(1 - (x0 - x)/x0)
    this is smaller than or equal to 2|(x0 - x)/x0| if |(x0 - x)/x0| < 1/2

    ok so first.. why is |(x0 - x)/x0| < 1/2?? did someone decide to randomnly pick 1/2...? and why is |(x0 - x)/x0| < 1/2... where did the ln go..? wouldnt it make more sense if it was .. ln|(x0 - x)/x0| < 1/2 ?

    Take delta = min (epsilon*(x0)/2 , 1/2*x0)

    how was delta chosen to be these??

    thank you! any help ... would save me a lot of time..effort. im looking through books..online.. believe me im trying but this is logic (and i am poor at it!)

    i havent written up the whole problem,only the stage i am stuck so far!
    This is hard to read and follow.

    You only need to prove that it's continuous at one and the rest follows. I don't know how you define the natural log though, like this \ln(x)=\int_1^x\frac{d<br />
\xi}{\xi}?
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  3. #3
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    Quote Originally Posted by matlabnoob View Post
    Before i posted this,i tried reading my notes again and again! so any help....id be thankful!
    here is the problem...

    consider f(x) = lnx, with A element in (0,1)
    confirm the continuity of f in (0,1)
    Let x0 be an element in (0,1)
    then |lnx - lnx0| = | ln x/x0 | = ln(1 - (x0 - x)/x0)
    this is smaller than or equal to 2|(x0 - x)/x0| if |(x0 - x)/x0| < 1/2

    ok so first.. why is |(x0 - x)/x0| < 1/2?? did someone decide to randomnly pick 1/2...? and why is |(x0 - x)/x0| < 1/2... where did the ln go..? wouldnt it make more sense if it was .. ln|(x0 - x)/x0| < 1/2 ?

    Take delta = min (epsilon*(x0)/2 , 1/2*x0)

    how was delta chosen to be these??

    thank you! any help ... would save me a lot of time..effort. im looking through books..online.. believe me im trying but this is logic (and i am poor at it!)

    i havent written up the whole problem,only the stage i am stuck so far!

    Do you mean that:

    δ =min{ \frac{\epsilon.x_{o}}{2} ,\frac{1}{2x_{o}}}
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  4. #4
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    Quote Originally Posted by xalk View Post
    Do you mean that:

    δ =min{ \frac{\epsilon.x_{o}}{2} ,\frac{1}{2x_{o}}}

    Yes this is what i mean!
    (how do you use mathematical symbols?i didnt know how to use them here)

    i am very very clueless about this problem still... and taking me hours to understand
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    Quote Originally Posted by Drexel28 View Post
    This is hard to read and follow.

    You only need to prove that it's continuous at one and the rest follows. I don't know how you define the natural log though, like this \ln(x)=\int_1^x\frac{d<br />
\xi}{\xi}?
    i understand the background of it..that i have to prove continuity and uniform continuity will imply continuity. but the problem confuses me so much. it looks all very random!
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by matlabnoob View Post
    Yes this is what i mean!
    (how do you use mathematical symbols?i didnt know how to use them here)

    i am very very clueless about this problem still... and taking me hours to understand
    Ok. Well you have three choices A) rewrite in LaTeX or a word document (with equation editor) and post it up and I will review it B) leave it as it is and maybe xalk or another member will help you C) I can give an alternative answer. Which do you want?
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  7. #7
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by matlabnoob View Post
    .that i have to prove continuity and uniform continuity will imply continuity
    I'm not sure what you mean by this. U.C. implies cont. but the other direction is not necessarily true (e.g. 0,1]\to\mathbb{R}" alt="f0,1]\to\mathbb{R}" />, with f:x\mapsto\frac{1}{x})
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  8. #8
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    Quote Originally Posted by Drexel28 View Post
    Ok. Well you have three choices A) rewrite in LaTeX or a word document (with equation editor) and post it up and I will review it B) leave it as it is and maybe xalk or another member will help you C) I can give an alternative answer. Which do you want?

    I hopethis turns out well...i tried in word(thanks!)


    Consider:
    f(x) = lnx, A Є (0,1)
    Confirm the continuity of f Є (0,1) by doing the following…
    Let x0 Є (0,1), then |lnx - ln x0| = |lnx / x0| = ln(1 – (x0 – x)/ x0| ≤ 2|( x0 – x)/ x0| if |( x0 – x)/ x0| < ˝
    Take δ = min {ε. x0/2, 1/2 x0}
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  9. #9
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    Quote Originally Posted by Drexel28 View Post
    I'm not sure what you mean by this. U.C. implies cont. but the other direction is not necessarily true (e.g. 0,1]\to\mathbb{R}" alt="f0,1]\to\mathbb{R}" />, with f:x\mapsto\frac{1}{x})
    aha..of course.thanks for this! will note down for future reference
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