Re: Question about Contiunuity Definition

On Nov 23, 10:15 am, Tonico <Tonic...@xxxxxxxxx> wrote:
On Nov 23, 6:58 am, Jack <rossb...@xxxxxxx> wrote:

Hi,

I'm using the definition of continuity that says:

let f be a mapping from R into R.

f is continuous at c if:

for every å>0 there exists a ä>0 s.t. |f(x)-f(c)| <å whenever |x-c|<ä and x is in R.

Alright, so my question is, are å and ä, specifically ä, in R as well, or do we not know. I figure it would be stupid if they aren't in R, but do I know for a fact that the ä that exists is in R, or do I have to show that it is/can be?

***************************************************************************-**********
Not only it is clear, imho, that epsilon and delta are reals... where
else could they be? I'm talking of real analysis, of course.

Now, in order to show f is continuous in c you have to prove the
existence of such a delta for any epsilon > 0 whatsoever, but the
"search" is done within R. In this respect you don't know for a fact
that such a delta exists, but you rather prove its existence.

Disclaimer: If you just want to get some answer for your assignment so
that you can get good grades, ignore the following. But if you want to
really think about this, read on.

Nope. those epsilon and delta do not exist in R. When you try to
verify them by thought process, they practically become infinitesimals
which are "excluded" from R. But you might reason that the specified
condition ensures that all points between c and x are covered. This is
only a belief based on the unknown properties of the infinite. So,
except by assuming that continuity is built by points, you can't prove
continuity using the standard R.

- venkat



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