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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
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