Characterizations of continuity
Dave L. Renfro
characterizations of continuity for functions from a metric
space to a (possibly different) metric space.
Characterizations of continuity
http://groups.google.com/group/sci.math/msg/2217151a43d678a1&fwc=1
A few minutes ago I adapted this post for a reply that
I made to the non-usenet group "ap-calculus" (archived
at The Math Forum), and I thought my comments were worth
archiving in usenet also. I'm not replying to my original
post above because google's deadline for replying has expired,
and so I'm going to begin a new thread with the same subject.
The following are well known to be equivalent, for a fixed
real-valued function f defined on a neighborhood of x = c,
and many of these are covered in most any undergraduate
real analysis text or advanced calculus text. Each of
these can be taken as the definition of "f is continuous
at x = c".
(1) (for all epsilon > 0)(there exists delta > 0)(for all x)
we have
|x - c| < delta ===> | f(x) - f(c) | < epsilon.
(2) (for all epsilon > 0)(there exists delta > 0)
such that
the image of {x: |x - c| < delta} under f is a subset of
{x: | x - f(c) | < epsilon}.
(3) ( for all open intervals J' that contain f(c) )
( there exists an open set U that contains c )
such that
the image of U under f is a subset of J'.
(3') In (3), replace "open set U that contains c"
with "open interval J that contains c".
(3") In (3), replace "open set U that contains c"
with "open interval J whose midpoint is c".
(4) ( for all open sets U' that contain f(c) )
( there exists an open set U that contains c )
such that
the image of U under f is a subset of U'.
(4') In (4), replace "open set U that contains c"
with "open interval J that contains c".
(4") In (4), replace "open set U that contains c"
with "open interval J whose midpoint is c".
(5) If x_n --> c, then f(x_n) --> f(c).
REMARKS: (3) says that, with regard to the condition
given in (4), we only need to verify it for those open
sets U' that are open intervals, and (2) implies that
we can further restrict ourselves to open intervals
whose centers are c. Also, (1) is simply a logical
reformulation of (2) (i.e. spelling out what "is a
subset of" means and what the image of a set under
a function means).
I've given 9 formulations, which I believe pretty much
exhausts all the standard variations, but there are
clearly others that could be used. For example, instead
of intervals centered at x = c, we could use intervals
(a,b) such that the ratio |c-a| : |b-c| is 1:2, and
similarly for intervals centered at f(c). Indeed,
these two types -- midpoint and 1:2 ratio -- could be
used simultaneously, one type for intervals containing
c and the other type for intervals containing f(c).
The concept of a filter base of sets includes all
of the variations on this last idea that anyone is
likely to come up with. For more information about
filters, see the following URLs, especially the
3'rd URL. By the way, the use of nets (a sequence
analog of filters) provides a way to interpret
"limit of Riemann sums over all partitions whose
norms approach zero" in a way that parallels the
usual idea of the limit of a sequence. An elementary
real analysis text that uses this approach is "Limits"
by Alan F. Beardon [Springer-Verlag, 1997]. For more
about the use of nets and filters in analysis, see
the 4'th and 5'th URLs below.
http://en.wikipedia.org/wiki/Filter_(mathematics)
http://www.efnet-math.org/w/Seminar_on_Filters_and_Topology
http://math.u-bourgogne.fr/topo/dolecki/Page/init_X.pdf
http://tinyurl.com/y7g36n
http://tinyurl.com/yl2nfq
Incidentally, many of the variations represented
by the 9 formulations I gave earlier have been
generalized to express all sorts of weaker notions
of "f is continuous at x = c". For instance, we could
replace the collection of intervals J' with a larger
collection of sets, such as any set that becomes an
interval with the addition of at most countably many
points, or we could restrict ourselves to sequences
converging to x = c that have a specified lower bound
for their "rapidity of convergence" and/or sequences
whose terms are restricted to certain sets (i.e. only
sequences with rational terms can be used). For many
of these weaker notions of continuity, the analogs
of the statements I gave above are not all equivalent.
For more about this, see the first three chapters of
"Real Functions" by Brian S. Thomson [Lecture Notes
in Mathematics #1170, Springer-Verlag, 1985].
Dave L. Renfro
Hero
> On September 18, 2006 I posted some pointwise and global
> characterizations of continuity for functions from a metric
> space to a (possibly different) metric space.
>
> Characterizations of continuity
> http://groups.google.com/group/sci.math/msg/2217151a43d678a1&fwc=1
>
....
>
> The following are well known to be equivalent, for a fixed
> real-valued function f defined on a neighborhood of x = c,
> and many of these are covered in most any undergraduate
> real analysis text or advanced calculus text. Each of
> these can be taken as the definition of "f is continuous
> at x = c".
>
> (1) ...
The base of continuity of a function is in the continuity of its
independent variable, here x, which varies through the reals.
The "deformation" of this variation is the way how the dependent
variable f ( x ) changes.
If it can keep this continuity of the reals ( it's denseness,
connectedness,..), than we call the function continuous, this can also
happen to be only partly, only in certain intervalls, or in a point and
it's surrounding.
With friendly greetings
Hero