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Topic: limit and continuity definition of real valued function of single
real variable

Replies: 9   Last Post: May 5, 2012 5:41 AM

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 Dave L. Renfro Posts: 4,442 Registered: 12/3/04
Re: limit and continuity definition of real valued function of single real variable
Posted: May 3, 2012 10:04 AM
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exmathematician wrote:

http://mathforum.org/kb/message.jspa?messageID=7810819

> Well, no, the stipulation that x be in the domain of f is
> incorrect. The point is that there must exist a delta such
> that the condition holds for ANY x satisfying 0 < |x-a| < delta.
> If there are values of x in that interval NOT in the domain
> of f, then a smaller delta will be necessary. If this is the
> case for ALL delta, then the condition is not satisfied.
> Special definitons may be made to deal with situations
> like sqrt(x) at 0, where the function is "continuous
> from the right".

If you look through several dozen real analysis texts in
any college/university library, you'll find that in almost
all of them the convention is to say something equivalent
to "for all x in dom(f) such that 0 < |x-a| < delta".
Note also that "x in dom(f)" is needed for consistency
with topological subspace formulations.

Dave L. Renfro

Date Subject Author
5/2/12 Lax
5/2/12 J. Antonio Perez M.
5/3/12 exmathematician
5/3/12 Dave L. Renfro
5/3/12 exmathematician
5/3/12 Artur Steiner
5/4/12 Kaba
5/4/12 Lax
5/5/12 Kaba
5/3/12 Ken.Pledger@vuw.ac.nz

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