Cotinuous Functions and Limits
Maury Barbato
I have two questionbs about continuity.
(I) Let f:[a,b]->R be a function such that the limit
lim_{x->x_0} f(x) exists for every x_0 in [a,b].
Does there exist a point x_0 where f is continuous?
This is not true if we replace [a,b] with a generic
subset X of R. Take, e.g., the function f:Q /\[0,1]->R,
defined as
f(x)=1/n, where x=m/n, with m and relatively prime.
(II) Let S and T be two topological spaces, and f:S->T
a function such that the limit lim_{x->x_0} f(x) exists
for every x_0 in S. Set, for every x_0 in S,
g(x_0)=lim_{x->x_0} f(x).
Is g a continuous function?
Thank you very very much for your help!
My Best Regards,
Maury
Dave L. Renfro
> (I) Let f:[a,b]->R be a function such that the limit
> lim_{x->x_0} f(x) exists for every x_0 in [a,b].
> Does there exist a point x_0 where f is continuous?
Any such function must be continuous at all but a
countable set of points. In fact, this continues to be
true even if you only assume, at each point, the existence
of at least one of the unilateral limits. [This last part
doesn't follow from what's below. However, similar methods
can be used to show that the set of left subsequence limit
points of a function f at a point y differs from the set
of right subsequence limit points of the function f at y
for at most countably many points y.] What follows is from
a 3 February 2003 post of mine:
For each n = 1, 2, 3, ... there can be at most countably many
values of y such that lim(x->y) f(x) = 0 and f(y) > 1/n. See if
you can prove this. [Hint: The set in question is isolated.] It
now follows that there can be at most countably many values of
y such that lim(x->y) f(x) = 0 and f(y) > 0. Thus, not only can
this not happen for all y in R, it isn't even possible for
uncountably many y's.
More generally, one can show that given any function f:R --> R
there are at most countably many points y for which the value f(y)
is not equal to lim(n --> oo) f(x_n) for some sequence x_n --> y
(i.e. f(y) is not equal to any of the subsequence limit points
of f at y).
Dave L. Renfro
Valeriu Anisiu
>
> > (I) Let f:[a,b]->R be a function such that the
> limit
> > lim_{x->x_0} f(x) exists for every x_0 in [a,b].
> > Does there exist a point x_0 where f is continuous?
>
> Any such function must be continuous at all but a
> countable set of points.
This follows from a result of the romanian mathematician
Alex Froda (1928) which asserts that the set of first kind discontinuities of any function f : [a,b] --> R is at most countable.
> In fact, this continues to be
> true even if you only assume, at each point, the
> existence
> of at least one of the unilateral limits.
This result was obtained by Wang Yim-Ming (1965)
for functions with values in a metric space.
>
> Dave L. Renfro
>
V. Anisiu
Dave L. Renfro
Dave L. Renfro wrote (in part):
>> Any such function must be continuous at all but a
>> countable set of points.
Valeriu Anisiu wrote (in part):
> This follows from a result of the romanian mathematician
> Alex Froda (1928) which asserts that the set of first kind
> discontinuities of any function f : [a,b] --> R is at
> most countable.
William H. Young established the result, indeed even the
stronger cluster set statements I gave, in 1907 or 1908,
but like much of Young's work, it was mostly overlooked
or forgotten. I think the specific result the original
poster asked about may have also been rediscovered by
Sierpinski around 1911-1914, but I'd have to look at
some notes and papers that I don't have with me to be
sure. Also, there are at least a couple of Amer. Math.
Monthly papers that rediscover the result as a way of
"strengthening" the Riemann integrability condition,
from almost everywhere continuity to almost everywhere
unilateral continuity (where the choice of side is not
required to be uniform throughout the interval).
Alex Froda proved a number of results that were of
the same general type obtained by Henry Blumberg
(various extensions and refinements of W. H. Young's
results in the cluster set properties of functions).
My impression is that Froda's work, even those results
first obtained by him, were even more overlooked and/or
forgotten than Young's or Blumberg's work in these areas,
although Blumberg at least was aware of them (because I
seem to recall that Blumberg was the Zbl reviewer for
several of Froda's papers).
I suspect in Froda's case, journal availability, his
relative isolation from much of the mathematical community,
and language issues (I know he knew French, because I
have several of his papers written in French, but he
may not have been able to read English papers even if
they were available) were why he wasn't very aware of
Young's and Blumberg's work.
Dave L. Renfro
Valeriu Anisiu
>
> Dave L. Renfro wrote (in part):
>
> >> Any such function must be continuous at all but a
> >> countable set of points.
>
> Valeriu Anisiu wrote (in part):
>
> > This follows from a result of the romanian
> mathematician
> > Alex Froda (1928) which asserts that the set of
> first kind
> > discontinuities of any function f : [a,b] --> R is
> at
> > most countable.
>
> William H. Young established the result, indeed even
> the
> stronger cluster set statements I gave, in 1907 or
> 1908,
> but like much of Young's work, it was mostly
> overlooked
> or forgotten.
..
>
> Dave L. Renfro
>
AFAIK, prior to Froda's result, the function f was
_assumed_ to have only first kind discontinuities.
Unfortunalely I did not checked the original papers
of Young but a reliable local source is very clear about this.
V. Anisiu
Dave L. Renfro
> AFAIK, prior to Froda's result, the function f was
> _assumed_ to have only first kind discontinuities.
> Unfortunalely I did not checked the original papers
> of Young but a reliable local source is very clear
> about this.
Young's paper is
[1] William H. Young, "On the distinction of right and
left at points of discontinuity", Quarterly Journal
of Pure and Applied Mathematics 39 (1907), 67-83.
[Dated June 1907.]
Just below Theorem 6 (p. 82) Young [1] states, using
present day notation, that given any function f:R --> R,
the following holds at all but countably many points y:
lim-inf(x --> y-) = lim-inf(x --> y+) <= f(y)
and
f(y) <= lim-sup(x --> y-) = lim-sup(x --> y+)
This implies that if f:R --> R has at least one unilateral
limit at each point (the choice of side is not required
to be uniform with respect to y), then f has at most
countably many discontinuities.
This theorem can also be found (with a proof) in Hobson's
"The Theory of Functions of a Real Variable", Volume 1,
3'rd edition, 1927. See Section 228: "The symmetry of
functional limits" on pp. 304-305.
The stronger result that the left cluster set of f(x)
at y equals the right cluster set of f(x) at y for
all but countably many points y was one of the topics
Young discussed in his 8 April 1908 talk at the 4'th
International Congress of Mathematicians in Rome.
The discussion is in Section 7 (p. 54) of the published
version ["On some applications of semi-continuous
functions", pp. 49-60].
In 1930 Henry Blumberg [2] proved a very general result
for functions of two real variables (the "functions"
can be many-valued, in fact) that, as a very special
case (Example 3, pp. 21-22) gives rise to Young's more
general cluster set result. The results of [2] are
discussed in [3] (pp. 822-825), and the special
application that gives rise to Young's cluster set
symmetry result can be found on the middle of p. 824.
[2] Henry Blumberg, "A theorem on arbitrary functions
of two variables with applications", Fundamenta
Mathematicae 16 (1930), 17-24. [Paper not dated.]
[3] Henry Blumberg, "Methods in point sets and the
theory of real functions", Bulletin of the American
Mathematical Society 36 (1930), 809-830.
[Invited talk given 18 April 1930.]
Dave L. Renfro
Valeriu Anisiu
..
>
> Dave L. Renfro
>
So, I am right, one cannot obtain Froda's result from Young's one. If f is the Dirichlet function (1 for rationals and 0 for irrationals), Young's inequalities simply reduce to 0 <= f <= 1 and says nothing about first kind discontinuities of f.
V. Anisiu
Dave L. Renfro
> So, I am right, one cannot obtain Froda's result from
> Young's one. If f is the Dirichlet function (1 for
> rationals and 0 for irrationals), Young's inequalities
> simply reduce to 0 <= f <= 1 and says nothing about
> first kind discontinuities of f.
I seem to be having a mental block. I thought "first kind
discontinuity" meant a jump discontinuity or a removable
discontinuity. Moreover, Dirichlet's function has an
uncountable number of these, so it's certainly not the
case that any function has only countably many first
kind discontinuities, unless the term means something
different than what I said. Or am I missing something?
Dave L. Renfro (a little confused now)
Valeriu Anisiu
Your definition of 1st kind discontinuity is OK, but the Dirichlet function does not have any such discontinuity
(at each point there is a disontinuity of the 2nd kind).
The problem is that I cannot see how Young' s inequalities imply that for any function the set of such discontinuities is at most countable.
V. Anisiu
Dave L. Renfro
>> I seem to be having a mental block. I thought "first
>> kind discontinuity" meant a jump discontinuity or a
>> removable discontinuity. Moreover, Dirichlet's function
>> has an uncountable number of these, so it's certainly not
>> the case that any function has only countably many first
>> kind discontinuities, unless the term means something
>> different than what I said. Or am I missing something?
Valeriu Anisiu wrote:
> Your definition of 1st kind discontinuity is OK, but the
> Dirichlet function does not have any such discontinuity
> (at each point there is a disontinuity of the 2nd kind).
> The problem is that I cannot see how Young' s inequalities
> imply that for any function the set of such discontinuities
> is at most countable.
Ooops, my mistake about the Dirichlet function.
As for Young's inequalities, I see your point now.
To review (in case others are reading), recall Young
proved that the following occurs at all but a countable
number of points y:
lim-inf(x --> y-) = lim-inf(x --> y+) <= f(y)
and
f(y) <= lim-sup(x --> y-) = lim-sup(x --> y+)
In order for me to write this all on one line so
that upcoming discussion is clearer, I'll use
the shorthand ('L' for 'lower' and 'U' for 'upper'):
lim-inf(x --> y-) = Lf(y-)
lim-inf(x --> y+) = Lf(y+)
lim-sup(x --> y-) = Uf(y-)
lim-sup(x --> y+) = Uf(y+)
Thus, at co-countably many points y we have
Lf(y-) = Lf(y+) <= f(y) <= Uf(y-) = Uf(y+).
Now, if at any of these co-countably many points y
we have Lf(y-) = Uf(y-) OR Lf(y+) = Uf(y+) (that is,
the left limit at y exists OR the right limit at y
exists), then we get equality for all 5 values. In
particular, if at EACH point y at least one of the
two unilateral limits exist (let alone both unilateral
limits, let alone the limit itself which was what
the original poster was asking us to assume), then
we have equality throughout on a co-countable set.
This certainly takes care of the original poster's
question and it's what I originally intended, but
it doesn't imply what you said, as far as I can
see now (and what you, I would suppose, have been
seeing throughout this discussion).
We want to show there are only countably many points
where Lf(y-) = Uf(y-) and Lf(y+) = Uf(y+) holds
without all 5 values being equal.
I'm not sure, but I think there is a result which
says that, for any function f, there exists a
co-countable set of points y such that f(y) belongs
to the intersection of C(f,y-) and C(f,y+). By C(f,y-),
I mean the left cluster set of f at the point y,
that is, the set of all left subsequence limits of f
at the point y. Similarly, C(f,y+) denotes the right
cluster set of f at the point y.
If this is true, then we would certainly have the
result you stated (and Alex Froda may have actually
proved this stronger result too, either initially
or in a later paper). Also, if this is true, then
Young may have proved it as well, but I'd want to
look through some of his papers when I'm at home
(which I'm not now) before claiming Young did this.
Dave L. Renfro
Maury Barbato
> Let S and T be two topological spaces, and
> f:S->T
> a function such that the limit lim_{x->x_0} f(x)
> exists
> for every x_0 in S. Set, for every x_0 in S,
> g(x_0)=lim_{x->x_0} f(x).
> Is g a continuous function?
>
This is surely true, if T has the following property
For every open subset A of T, and every point a in A,
there is an open subset B, containing a, such that Cl(B)
is contained in A.
But this property doesn't hold in a generic topological
space, so ...
Maury Barbato
Maury Barbato
> Maury wrote:
>
> > Let S and T be two topological spaces, and
> > f:S->T
> > a function such that the limit lim_{x->x_0} f(x)
> > exists
> > for every x_0 in S. Set, for every x_0 in S,
> > g(x_0)=lim_{x->x_0} f(x).
> > Is g a continuous function?
> >
>
> This is surely true, if T has the following property
>
> For every open subset A of T, and every point a in
> A,
> there is an open subset B, containing a, such that
> Cl(B)
> is contained in A
>
> (it's easy to prove using the defintion of limit).
>
> But this property doesn't hold in a generic
> topological
> space, so ...
>
> Maury Barbato
No, my conjecture is not generally true.
Take T={a,b}, with the topology {{a},{a,b},Ø}.
Let f:[0,1]->T be the function f(x)=a for every x.
Set
b if 0<=x<1/2
g(x)=
a if 1/2<=x<=1
g is not continuous!!!
Surely T is not a Hausdorff space: I don't know if
my conjecture holds when T is supposed to be Hausdorff.
My Best Regards,
Maury Barbato
Dave L. Renfro
http://groups.google.com/group/sci.math/msg/b6d5cc548dea44e1
> Also, if this is true, then Young may have proved it
> as well, but I'd want to look through some of his papers
> when I'm at home (which I'm not now) before claiming
> Young did this.
After checking some references at home last night, it seems
Young did prove a result, published in 1908, that implies an
arbitrary function has at most countably many discontinuities
of the first kind.
Let C(f,x-) and C(f,x+) denote the left and right cluster
sets of the function f at the point x. In the following
paper Young proved that, at a co-countable set of points x,
we have C(f,x-) = C(f,x+) and f(x) belongs to these sets.
William H. Young, "Sulle due funzioni a più valori costituite dai
limiti d'una variabile reale a destra e a sinistra di ciascun punto"
[On the two many-valued functions obtained from the right and left
limits of a function of one variable at every point ], Atti della
Reale Accademia dei Lincei. Rendiconti, classe di scienze fisiche,
matematiche e naturali (5) 17(1) (1908), 582-587. [JFM 39.0469.03]
Incidentally, there are some differences in the title in some
sources I looked at, and I don't have a copy of the paper in
question, so I'm not exactly sure of the title. For example,
in Young's "On the discontinuities of a function of one or more
real variables" [Proc. London Math. Soc. (2) 8 (1910), 117-124],
a paper that considers generalizations of the result to functions
of several variables (the exceptional sets do not have to be
countable, but they do form a certain subclass of sets simultaneously
measure zero and first category -- even a certain subclass of
the sigma-cone porous sets I believe), the title is given as
"Sulle due funzioni a più valori costituite dai limiti d'una
funzione d'una variabile a destra e a sinistra di ciascun punto".
Then, in G. C. Young's "On the Derivates of a Function" [Proc.
London Math. Soc. (2) 15 (1917), 360-384], the title is the
same as the JFM review's title (which is the title I gave
above), except that "limite" (singular) is used instead of
"limiti" (plural). Finally, the title in Bruckner/Thomson's
"Real variable contributions of G. C. Young and W. H. Young"
[Expositions Mathematicae 19 (2001), 337-358] gives the same
title as in the on-line JFM review, except for the typo
"funzionia più" in place of "funzioni a più".
Bruckner/Thomson state and briefly discuss the Young's result
I gave above on p. 345. They cite a different paper by Young,
but I think the paper they cite is only for the "f(x) belongs
to" part of the theorem. I don't know for sure if this part
is in the Young paper I cited, but I do know that the cluster
set equality part (which I've previously posted was announced
by Young at the 1908 International Congress of Mathematicians
in Rome) is proved in the paper I cited, and "f(x) belongs
to [lim-inf f, lim-sup f]" is in the paper that Bruckner/Thomson
cite ["A theorem in the theory of functions of a real variable",
Rendiconti del Circolo Matematico di Palermo 24 (1907), 187-192].
Dave L. Renfro
Dave L. Renfro
> After checking some references at home last night, it seems
> Young did prove a result, published in 1908, that implies an
> arbitrary function has at most countably many discontinuities
> of the first kind.
I meant to add, which I had already pointed out yesterday but
I wanted to make sure this is clear, that Valeriu Anisiu was
correct in saying that the specific results I had cited earlier
did not imply an arbitrary function has at most countably
many discontinuities of the first kind. What was missing was
the part where we have f(x) \in [lim-inf f, lim-sup f] at
co-countably many points x.
Dave L. Renfro
Valeriu Anisiu
> space.
>
> Let X = { (x,y) | y>=0, x,y rationals}, and fix an
> irrational numbert.The irrational slope topology T on
> X
> is generated by epsilon-nbds ofthe form N_e( (x,y) )
> =
> { (x,y) } \/ B_e(x + (y/t)) \/ B_e(x - (y/t))where
> B_e(z) = { r in Q | |r-z|<e } where Q is the
> rationals
> on the x axis.So each N_e( (x,y) ) consists of (x,y)
> and
> two intervals on therational x-axis, centered at the
>
> irrational points x + (y/t) and x - (y/t). The lines
>
> that join these points to (x,y) have slope t
> and-t.(X,T)
> is Hausdorff: since t is irrational, no two points of
> X
> can lieon a line with slope t, and if one point of X
>
> lies on a line withslope t, then no other point of X
> can
> line on the line with slope -twhich intersects the
> original line at its intersection with the xaxis. So
> any
> two distinct points in X project (along the lines
> ofslope t and -t) onto distinct pairs of irrational
> points on thex-axis, and these have disjoint
> neighborhoods.The closure of each basis nbd N_e(
> (x,y) )
> contains the union of thefour strips of slope t and
> -t
> that emanate from B_e(x + (y/t) ) andB_e(x - (y/t) ),
>
> since every point in such a ray projects to
> anirrational
> on the x axis that lies within e of either x+(y/t)
> or
> x-(y/t).
>
> Now set
> S=(((1/t)-1,(1/t)+1)/\Q)\/{(1/t)+(1/(2^n)):n in N,
> n>=1}
>
> and define the function f:S->T as follows
>
> (x,0) if x is in S/\Q
> f(x)=
> (0,0) otherwise
>
> It's quite easy to verify that g is not
> continuous!!!
>
> My Best Regards,
> Maury
If T is regular (i.e. each point has a fundamental system
of closed neighborhoods)) then the property holds
(as you noticed).
I suspect that the regularity is also necessary in order
to have the property for each S and f.
Note also that T should be Hausdorff because otherwise
lim_{x->x_0} f(x) is not a proper notation (the limit may
be not unique).
V. Anisiu
Maury Barbato
Ops, a trivial typo, f(x) is defined as follows
(x,0) if x is in S/\Q
f(x)=
(0,1) otherwise
Maury Barbato
> If T is regular (i.e. each point has a fundamental
> system
> of closed neighborhoods)) then the property holds
> (as you noticed).
> I suspect that the regularity is also necessary in
> order
> to have the property for each S and f.
I'm quite sure this is true, but my scanty familiarity
with topology makes the prove too difficult for me!!
> Note also that T should be Hausdorff because
> otherwise
> lim_{x->x_0} f(x) is not a proper notation (the limit
> may
> be not unique).
>
> V. Anisiu
My Best Regards,
Maury