Re: Attempt epsilon-delta

From: Chris Wagner (clwagner_at_vulcan.wagner.nul)
Date: 10/01/04 

Date: Fri, 1 Oct 2004 12:25:14 +0000 (UTC)

In article <cjevb0$1q9h$1@agate.berkeley.edu>,
        magidin@math.berkeley.edu (Arturo Magidin) writes:
> In article <cjeohn$3cvr$1@murrow.it.wsu.edu>,
> Chris Wagner <clwagner@vulcan.wagner.nul> wrote:
>>Greetings,
>>
>>Here is my attempt at an elementary epsilon-delta proof.
>>
>>Prove lim_{x->a} f(x) = f(a), for continuous f(x) in Reals.
>
> What is your definition of continuity and of limits? I ask, because
> sometimes "continuous" is defined to mean "the function is defined at
> the point and the value of the limit exists and agrees with the value
> of the function", which would clearly make this a tautological
> statement.

It did seem a bit tautological to me. My purpose was to simplify so I
could understand epsilon-delta well enough to construct proofs. It is
my understanding that the Cauchy-Weierstrass epsilon-delta method is
considered definitive, that is, if I can correctly do e-d I can Prove
claims about limits.

>
> I assume that you are using the following definitions for limit and
> for "continuity at x=a":
>
> DEF. Let f(x) be a function. The limit as x goes to a of f(x) is equal
> to L, lim_{x->a} f(x) = L if and only if
>
> for every e>0 there exists d>0 such that
>
> if 0< | x-a | < d, then |f(x)-L|< e.
>
>
> DEF. Let f(x) be a function. Then f(x) is continuous at a if and only
> if f is defined at a, and
>
> for every e>0 there exists d>0 such that
>
> if |x-a| < d, then |f(x)-f(a)|<e.
>

These are indeed the e-d definitions I hope to apply. I read the
definitions in Rudin's "Principles of Mathematical Analysis" (third
ed.), begining of chapt 4.

Perhaps I can state my problem directly from the two definitions
quoted above:

Given continuous function f(x) and e>0 what is d? I need to
understand how to construct the d (my response to your e challenge),
since that seems to be the essence of doing such proofs.

>>Let
>>epsilon>0 be given. Using the Mean Value Theorem (MVT)
>
> I'm the one who usually uses cannonballs to swat flies... but isn't
> this a bit beyond that?

Urk-gulp.

[snip]
>
> Look at the definitions above. Can you not see why if the second one
> holds, then the first one will hold with L=f(a)?

OK, if the second definition holds then we know there is a delta. But
what is it? (d as function of f and e)

>
[snip]
>>Is my attempt acceptable or salvageable?
>
> I would consider it completely unacceptable.
>

Thank-You for your reply to my post.

[snip]
>>Does not continuity mean lim_{x->a}f(x)=f(a),
>
> Depends on the definitions.
>
>> without requiring
>>inverse f or differentiability of f assumption?
>
> Correct. Look at the definitions you have.
>

OK, Good.

Thanks,
Chris Wagner

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