1. ## 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!

2. Originally Posted by matlabnoob
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 $\displaystyle \ln(x)=\int_1^x\frac{d \xi}{\xi}$?

3. Originally Posted by matlabnoob
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{$\displaystyle \frac{\epsilon.x_{o}}{2} ,\frac{1}{2x_{o}}$}

4. Originally Posted by xalk
Do you mean that:

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

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

5. Originally Posted by Drexel28
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 $\displaystyle \ln(x)=\int_1^x\frac{d \xi}{\xi}$?
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!

6. Originally Posted by matlabnoob
Yes this is what i mean!
(how do you use mathematical symbols?i didnt know how to use them here)

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?

7. Originally Posted by matlabnoob
.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. $\displaystyle f0,1]\to\mathbb{R}$, with $\displaystyle f:x\mapsto\frac{1}{x}$)

8. Originally Posted by Drexel28
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}

9. Originally Posted by Drexel28
I'm not sure what you mean by this. U.C. implies cont. but the other direction is not necessarily true (e.g. $\displaystyle f0,1]\to\mathbb{R}$, with $\displaystyle f:x\mapsto\frac{1}{x}$)
aha..of course.thanks for this! will note down for future reference