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Topic: Nonstandard Analysis Continuity Question
Replies: 14   Last Post: Oct 2, 2012 3:04 PM

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G. A. Edgar

Posts: 2,493
Registered: 12/8/04
Re: Nonstandard Analysis Continuity Question
Posted: Oct 1, 2012 9:01 AM
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In article
<547d1b5c-aaaa-4c2c-9e97-f28c86a6a724@x14g2000yqh.googlegroups.com>,
<nonstandardanalysis@yahoo.com> wrote:

> On Sep 29, 7:46 pm, FredJeffries <fredjeffr...@gmail.com> wrote:
> > On Sep 27, 8:04 pm, nonstandardanaly...@yahoo.com wrote:
> >

> > > Suppose we have a function f: *R -> { 0, 1 } defined as f(x) = 1 if x
> > > is limited, and f(x) = 0 if x is unlimited.  Is f continuous?  On one
> > > hand it doesn't appear that f is continuous since the function jumps
> > > from 0 to 1 without meeting any of the points in between.  However,
> > > there is no point of discontinuity.  Since f is an external function,
> > > S-continuity doesn't apply.

> >
> > The discontinuity does not occur at a point. It occurs at a gap, like
> > the function from the rationals to {0, 1} defined as
> > g(x) = 1 if x^2 < 2 and g(x) = 0 if x^2 > 2

>
> It didn't occur to me that a function could be continuous at every
> point, but still not be continuous overall.


And what do you mean by "continuous overall"???

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/



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