Semicontinuous but almost everywhere discontinuous?
Snis Pilbor
I'm trying to think of a function R->R which is semicontinuous
(lower or upper, of course the difference is superficial) which is
discontinuous almost everywhere.
It seems to be an extremely difficult problem. We have to
introduce lots of jump discontinuities to make the function
discontinuous almost everywhere, but then if we draw a line between the
values at a jump discontinuity (ie a line of height 1/2 when the
function is the characteristic function of the rationals) and ask whats
the inverse image of the stuff above that line, we inevitably get a
nonopen set (in this example the rationals).
Can anyone throw a hint my way? Yes, this is homework related.
Thanks,
S.P.
Dave L. Renfro
> I'm trying to think of a function R->R which is semicontinuous
> (lower or upper, of course the difference is superficial) which
> is discontinuous almost everywhere.
[snip]
> Can anyone throw a hint my way? Yes, this is homework related.
Here's a hint for a more precise result. The parts of this
hint are things that, in my opinion, should more or less
have been given as hints for your problem.
THEOREM: If E is a meager (i.e. first category) F_sigma set,
then there exists a semicontinuous function f
such that the discontinuity set of f equals E.
PROOF:
1. Show that if an increasing sequence of continuous functions
converges pointwise to a function f, then f is lower
semicontinuous. [In fact, this can be simultaneously
strengthened in two ways: The pointwise supremum of an
arbitrary collection of lower semicontinuous functions,
when this supremum is finite at each point, is a lower
semicontinuous function.]
2. Show that E = C_1 union C_2 union C_3 union ..., where
each C_n is closed and nowhere dense.
3. Let f_n(x) = 1 if x belongs to C_n and f_n(x) = 0 if
x does not belong to C_n. Show that each f_n is
continuous everywhere.
4. Let f = (1/2)*f_1 + (1/4)*f_2 + ... + (1/2^n)*f_n + ...
Show that f(x) exists for each real number x, f is
discontinuous at each x in E, and (optional, since
this part isn't needed for your problem) f is continuous
at each x not in E.
5. Explain why f is lower semicontinuous.
6. Explain why there exists a meager F_sigma set whose
complement has measure zero. (The March 22 sci.math
thread "Need example of first category subset" will
help, if you need it.)
REMARK: The discontinuity set of any lower semicontinuous
function (indeed, of any Baire one function) is a
meager F_sigma set. Hence, the discontinuity set of
a lower semicontinuous function can be characterized
by the property of being a meager F_sigma set. The
same characterization holds for upper semicontinuous
functions as well. [Use the fact that f is lower
semicontinuous if and only if -f is upper semicontinuous.]
Dave L. Renfro
The World Wide Wade
<1145232166....@j33g2000cwa.googlegroups.com>,
"Snis Pilbor" <snisp...@yahoo.com> wrote:
If E is a closed nowhere dense subset of R, set f(x) =
sin^2(1/d(x,E)) for x in R\E, f = 0 on E. Then f is continuous on
R\E, discontinuous at each point of E, and lower semicontinuous
on R. A function of the form sum(n=1,oo) f_n/2^n, where the f_n's
are defined relative to certain sets En as above, should do the
job.
The World Wide Wade
<1145237266.4...@t31g2000cwb.googlegroups.com>,
"Dave L. Renfro" <renf...@cmich.edu> wrote:
> Snis Pilbor wrote (in part):
>
> > I'm trying to think of a function R->R which is semicontinuous
> > (lower or upper, of course the difference is superficial) which
> > is discontinuous almost everywhere.
>
> [snip]
>
> > Can anyone throw a hint my way? Yes, this is homework related.
>
> Here's a hint for a more precise result. The parts of this
> hint are things that, in my opinion, should more or less
> have been given as hints for your problem.
>
> THEOREM: If E is a meager (i.e. first category) F_sigma set,
> then there exists a semicontinuous function f
> such that the discontinuity set of f equals E.
>
> PROOF:
>
> 1. Show that if an increasing sequence of continuous functions
> converges pointwise to a function f, then f is lower
> semicontinuous. [In fact, this can be simultaneously
> strengthened in two ways: The pointwise supremum of an
> arbitrary collection of lower semicontinuous functions,
> when this supremum is finite at each point, is a lower
> semicontinuous function.]
I think you need the remark in brackets for the below to work.
> 2. Show that E = C_1 union C_2 union C_3 union ..., where
> each C_n is closed and nowhere dense.
Isn't that just the definition of first category?
> 3. Let f_n(x) = 1 if x belongs to C_n and f_n(x) = 0 if
> x does not belong to C_n. Show that each f_n is
> continuous everywhere.
No, your f_n is discontinuous everywhere on C_n. I also think you
want f_n(x) = 0 if x belongs to C_n and f_n(x) = 1 if x does not
belong to C_n. That will give you a lower semicontinous function
on R.
> 4. Let f = (1/2)*f_1 + (1/4)*f_2 + ... + (1/2^n)*f_n + ...
> Show that f(x) exists for each real number x, f is
> discontinuous at each x in E,
Don't you want the C_n's to be pwdj for this argument?
Dave L. Renfro
>> THEOREM: If E is a meager (i.e. first category) F_sigma set,
>> then there exists a semicontinuous function f
>> such that the discontinuity set of f equals E.
>>
>> PROOF:
>>
>> 1. Show that if an increasing sequence of continuous functions
>> converges pointwise to a function f, then f is lower
>> semicontinuous. [In fact, this can be simultaneously
>> strengthened in two ways: The pointwise supremum of an
>> arbitrary collection of lower semicontinuous functions,
>> when this supremum is finite at each point, is a lower
>> semicontinuous function.]
The World Wide Wade wrote:
> I think you need the remark in brackets for the below to work.
Yes, I think so too!
Dave L. Renfro wrote:
>> 2. Show that E = C_1 union C_2 union C_3 union ..., where
>> each C_n is closed and nowhere dense.
The World Wide Wade wrote:
> Isn't that just the definition of first category?
It's easy, but not automatic in the sense that you might be
thinking. For example, an F_sigma sigma-porous set does not
have to be expressible as a countable union of closed porous
sets [1]. The key with F_sigma meager sets is that a closed
meager set is automatically nowhere dense.
[1] An example is given in Miroslav Zeleny, "The Banach-Mazur
game and sigma-porosity", Fund. Math. 150 (1996), 197-210.
This non-commutativity between (more generally) "contained
in an F_sigma sigma-small set" and "contained in a countable
union of closed small sets" is discussed on p. 575 of my
paper "Porosity, nowhere dense sets and a theorem of Denjoy",
Real Analysis Exchange 21 (1995), 572-581. Incidentally, this
distinction holds for sigma-upper-porosity (porosity defined
by a lim-sup), but not for sigma-lower-porosity (porosity
defined by a lim-inf).
Dave L. Renfro wrote:
>> 3. Let f_n(x) = 1 if x belongs to C_n and f_n(x) = 0 if
>> x does not belong to C_n. Show that each f_n is
>> continuous everywhere.
The World Wide Wade wrote:
> No, your f_n is discontinuous everywhere on C_n. I also think you
> want f_n(x) = 0 if x belongs to C_n and f_n(x) = 1 if x does not
> belong to C_n. That will give you a lower semicontinous function
> on R.
Huge ooops!! The characteristic function of a *closed* set is
*upper* semicontinuous and the characteristic function of an
*open* set is *lower* semicontinuous. But, worse than this, you're
exactly correct -- my f_n's are hardly continuous! I believe I
was thinking of restriction functions and then being able to
extend to the real line (Why? I was debating whether to include
the variations I brought up in <http://tinyurl.com/j6jac>, which
I decided not to in the end, and lost track of what I was doing.),
but of course this has almost nothing to do with what's at hand.
Dave L. Renfro wrote:
>> 4. Let f = (1/2)*f_1 + (1/4)*f_2 + ... + (1/2^n)*f_n + ...
>> Show that f(x) exists for each real number x, f is
>> discontinuous at each x in E,
The World Wide Wade wrote:
> Don't you want the C_n's to be pwdj for this argument?
(pwdj -- pairwise disjoint) I think this part goes through
without disjointness of the C_n's, but I don't have the time
right now to think about it, especially given the errors above
(which would want me to think over things a bit before rushing
off a reply like I did last night).
Dave L. Renfro
Dave L. Renfro
> This non-commutativity between (more generally) "contained
> in an F_sigma sigma-small set" and "contained in a countable
> union of closed small sets" [...]
More precisely, "non-commutativity" above should be "distinction".
The non-commutativity is (essentially) between the operation
of forming countable unions of closed sets and the operation
of forming countable unions of small sets.
These operations are commutative for meager sets (and result
in a collection of sets that are cofinal, in the sense of
inclusion, in the collection of meager sets) and they are
commutative for Lebesgue measure zero sets (and result a
collection of sets that are NOT cofinal, in the sense of
inclusion, in the collection of Lebesgue measure zero sets).
However, as I mentioned previously, they are not commutative
for the collection of sigma-upper-porous sets (porosity defined
by using a lim-sup). In fact, in the case of sigma-porous sets,
each of the following sigma-ideals is strictly contained in
the next sigma-ideal: subsets of countable unions of closed
porous sets, subsets of F_sigma sigma-porous sets, (subsets of)
sigma-porous sets.
Dave L. Renfro
The World Wide Wade
<1145278820.1...@v46g2000cwv.googlegroups.com>,
"Dave L. Renfro" <renf...@cmich.edu> wrote:
Here's what I was thinking. Let's use characteristic functions
like you were doing (there's no need for sin(1/...) as in my
previous post). Find closed nowhere dense Cn's that are pwdj such
that m(R\(UC_n)) = 0. Define f_n = 0 on C_n, f_n = 1/2^n on
R\C_n. Then f_n is discontinuous on C_n, continuous on R\C_n, and
lsc on R. Our hero is f = sum(n=1,oo) f_n, which is lsc on R. We
want to show f is discontinuous everywhere on U C_n. Fix n and
let x be in C_n. Then f = f_n + all the rest. But the sum
defining all the rest converges uniformly on R, hence is
continuous on C_n, simply because each summand is; that's where
pwdj comes in! It follows that f is discontinuous on C_n.