Continuity in each variable vs. joint continuity

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Dave L. Renfro

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Jun 4, 2005, 3:57:44 PM6/4/05
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The following two facts often arise in sci.math:

** The fact that a function f(x,y) can be continuous
in each variable separately, but not be continuous.

** The fact that a function f(x,y) can be continuous
relative to every line through a given point but
not be continuous at that point.

For example, within the past week, two threads have
appeared on these issues -- "limit existence of two
variable function" and "Multivariate functions limits".

For this reason, and because these issues came up
in some papers I was organizing and filing away today,
I thought I'd post a short essay on this topic.

Let g(x) = exp(-1/x^2), with g(0) left undefined.

Define f: R^2 --> R by

f(x,y) = 1 on the graph of y = g(x)
f(x,y) = 0 otherwise

Then f is continuous along every real-analytic curve
passing through the origin, but f is not continuous
at the origin. This example is given in footnote 3
on p. 31 of the following paper:

Arthur Rosenthal, "On the continuity of functions of
several variables", Mathematische Zeitschrift 63
(1955), 31-38. [Zbl 64.30003; MR 17,245c]
http://www.emis.de/cgi-bin/Zarchive?an=0064.30003

Rosenthal's paper is on-line at
http://dz-srv1.sub.uni-goettingen.de/cache/toc/D158728.html

Rosenthal established reasonably sharp upper and
lower bounds on results of this type by proving
the following theorem:

Theorem 1 (p. 31): If the single-valued function f is
continuous at p_0 along every convex curve through p_0
which is (at least) once differentiable, then f is also
continuous at p_0 as a function of (x,y). Yet f can
be continuous along every curve through p_0 which is
(at least) twice differentiable without being continuous
at p_0 as a function of (x,y).

The next paper gives a nice elementary and historical
survey of this topic. (Rosenthal's example is given and
cited on p. 123.)

Zbigniew Piotrowski, "The genesis of separate versus joint
continuity", Tatra Mountains Mathematical Publications
8 (1996), 113-126. [MR 98j:01026; Zbl 914.01007]
http://www.emis.de/cgi-bin/MATH-item?0914.01007
http://tatra.mat.savba.sk/paper.php?id_paper=357

Piotrowski's paper is on-line at
http://cc.ysu.edu/~zpiotrow/newpub.htm

I also have a scanned .pdf file for Piotrowski's paper
if anyone is interested. (For some reason I can't get
the on-line version to compile.)

To more efficiently state a few results that may be
of interest, I'll use the following terminology.

We say that f(x,y) is "coordinate-wise continuous" if

(a) for each fixed x_0, f(x_0,y) is continuous
and
(b) for each fixed y_0, f(x,y_0) is continuous.

We say that f(x,y) is "linearly continuous" if the
restriction of f to every line in the plane is
continuous.

We say that a set is "c-dense" if its intersection
with every open set has cardinality c. "Co-meager"
means the complement of a set of first Baire category
(i.e. the complement is a countable union of nowhere
dense sets).

René Baire proved in his 1899 Ph.D. Dissertation
[1, p. 27] that if f(x,y) is coordinate-wise continuous,
then f is continuous at each point of a certain set
that is c-dense in R^2.

More precisely, Baire proved that if f(x,y) is
coordinate-wise continuous, then the set of continuity
points of f contains a co-meager set of horizontal
lines and a co-meager set of vertical lines. In
particular, f is continuous at co-meagerly many
points of R^2. By the way, this last statement
follows easily from the statement before it, being
the easy converse of the Kuratowski-Ulam theorem.
It also follows from the next result.

Baire also proved in his Dissertation that every
coordinate-wise continuous function f(x,y) is a
Baire-one function. That is, f is a pointwise limit
of a sequence of continuous functions from R^2 to R.
See [2] for a short explicit proof of this result.

In 1910, William H. Young and Grace C. Young [3]
constructed a linearly continuous function f(x,y)
that is discontinuous at each point of a certain
set that is c-dense in R^2. Richard Kershner remarks
[4, p. 83] that Young/Young's example can easily be
modified to give a function f(x,y) that is continuous
relative to every analytic arc and yet is discontinuous
at each point of a certain set that is c-dense in R^2.
I suspect the same example could be modified using
Rosenthal's example to obtain a function continuous
relative to even more subsets of R^2 while still being
discontinuous at each point of some set that is c-dense
in R^2, but I am not aware of any such result that
has been published.

G. Tolstov [5] proved in 1949 that linearly continuous
functions are determined by their values on dense sets.
A much shorter and more elementary proof of this was
given by Casper Goffman and C. J. Neugebauer in 1961 [6].

[1] http://www.emis.de/cgi-bin/JFM-item?30.0359.01

[2] http://www.emis.de/cgi-bin/MATH-item?0208.07501

[3] http://www.emis.de/cgi-bin/JFM-item?40.0437.03

[4] http://www.emis.de/cgi-bin/MATH-item?0063.09017

[5] http://www.emis.de/cgi-bin/Zarchive?an=0038.04003
English translation: American Mathematical Society
Translation 1952, no. 69, 30 pages. (MR 13,926a)

[6] http://www.emis.de/cgi-bin/Zarchive?an=0101.04603

I'll end with an example of a coordinate-wise continuous
function whose discontinuity set is any specified
countable dense subset of R^2. This example is given
on p. 123 of Piotrowski's paper, and it is a slight
variation of a construction given on p. 34 of Rosenthal's
paper.

Let D = {(x_n, y_n) : n = 1, 2, 3, ...} be a countable
dense subset of R^2.

Let g_n(x,y) = 2(x - x_n)(y - y_n) / A(x,y)_n,

where A(x,y)_n = (x - x_n)^2 + (y - y_n)^2.

Let f_n(x,y) = 0 if (x,y) = (x_n, y_n)
and f_n(x,y) = g_n(x,y) otherwise.

Finally, define f(x,y) = sum(n=1 to infinity)
of f_n(x,y) / 2^n.

Then f is coordinate-wise continuous and the discontinuity
set of f is D (proof left to the reader).

Dave L. Renfro

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