Yet another counterexample challenge!

  • #1
22,097
3,279

Main Question or Discussion Point

Well, the last thread of counterexamples was pretty fun. So why not do it again! Again, I present you a list with 10 mathematical statements. The only rub now is that only ##9## are false, thus one of the statements is true. Provide a counterexample to the false statements and a proof for the correct one!

Rules:
  • For a counterexample to count, the answer must not only be correct, but a detailed argumentation must also be given as to why it is a counterexample.
  • Any use of outside sources is allowed, but do not look up the question directly. For example, it is ok to go check analysis books, but it is not allowed to google the exact question.
  • If you previously encountered this statement and remember the solution, then you cannot participate in this particular statement.
  • All mathematical methods are allowed.

Here you go:

  1. SOLVED BY PeroK ##\mathbb{R}## is the disjoint countable union of closed intervals
  2. SOLVED BY mfb Any open set in ##\mathbb{R}^2## is the disjoint countable union of open balls, where an open ball is a set of the form ##B(\mathbf{a},r) = \{\mathbf{x}\in \mathbb{R}^2~\vert~(a_1 - x_1)^2 + (a_2 - x_2)^2 < r^2\}## for ##\mathbf{a}\in \mathbb{R}^2## and ##r>0##.
  3. SOLVED BY andrewkirk Let ##\pi:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}## be a bijection and let ##\sum_{n} a_n## be an absolutely convergent series in ##\mathbb{R}##. Then ##\sum_n a_n = \sum_{k=0}^\infty \sum_{l=0}^\infty a_{\pi(k,l)}##.
  4. SOLVED BY PeroK Every continuous bounded function ##\mathbb{R}\rightarrow \mathbb{R}## is uniformly continuous.
  5. SOLVED BY Samy_A Every function ##\mathbb{R}\rightarrow \mathbb{R}## is the derivative of some function.
  6. SOLVED BY PeroK If ##f:\mathbb{R}\rightarrow \mathbb{R}## is continuous and sends open sets to open sets, then it also sends closed sets to closed sets.
  7. SOLVED BY mfbThere is no monotonic function ##\mathbb{R}\rightarrow \mathbb{R}## whose set of discontinuities is ##\mathbb{Q}##.
  8. SOLVED BY Samy_A Every nonconstant function ##\mathbb{R}\rightarrow \mathbb{R}## that is periodic has a smallest period.
  9. SOLVED BY andrewkirk For any infinitely differentiable monotonic function ##f:\mathbb{R}\rightarrow \mathbb{R}## such that ##\lim_{x\rightarrow +\infty} f(x) = 0##, also holds that ##\lim_{x\rightarrow +\infty} f^\prime(x) = 0##.
  10. SOLVED BY andrewkirk Every subset of ##[0,1]## of measure zero is the set of points of discontinuity of a Riemann-integrable function ##[0,1]\rightarrow \mathbb{R}##.

Thank you all for participating! I hope some of these statements were surprising to some of you and I hope some of you have fun with this! Don't hesitate to post any feedback in the thread!
 
Last edited:

Answers and Replies

  • #2
Samy_A
Science Advisor
Homework Helper
1,241
510
Every nonconstant function ##\mathbb{R}\rightarrow \mathbb{R}## that is periodic has a smallest period.
Define ##f:\mathbb{R}\rightarrow \mathbb{R}## as follows
##\left\{\begin{array}{l}
x \in \mathbb Q : x \mapsto 1\\
x \notin \mathbb Q : x \mapsto 2
\end{array}\right.##
Every rational number is a period, since for every rational number ##q##, ##x+q## will be rational if and only if ##x## is rational. That implies that for every rational number ##q##, ##\forall x \in \mathbb R: f(q+x)=f(x)##.
 
  • #3
Samy_A
Science Advisor
Homework Helper
1,241
510
Every bounded function ##\mathbb{R}\rightarrow \mathbb{R}## is uniformly continuous.
That seems trivial. Did you mean continuous bounded function?
 
  • #4
22,097
3,279
That seems trivial. Did you mean continuous bounded function?
Yes, sorry!
 
  • #5
Samy_A
Science Advisor
Homework Helper
1,241
510
Every function ##\mathbb{R}\rightarrow \mathbb{R}## is the derivative of some function.
I feel there must be an easier one, but this should work.
The derivative of a function must be Lebesgue measurable.

Proof: assume ##f=g'##
That means that ##\displaystyle \forall x \in \mathbb R: \lim_{h \rightarrow 0} \frac{g(x+h)-g(x)}{h}=g'(x)=f(x)##.
Define ##f_n:\mathbb{R}\rightarrow \mathbb{R}: x \mapsto \frac{g(x+1/n)-g(x)}{1/n}##
Then ##\displaystyle \forall x \in \mathbb R: \lim_{n \rightarrow +\infty}f_n(x)=g'(x)=f(x)##.
Since ##g## is differentiable, it is also Lebesgue measurable. Hence the functions ##f_n## are also Lebesgue measurable.
##f## being the pointwise limit of Lebesgue measurable function is itself Lebesgue measurable.

So a counterexample is a function ##f## that is not Lebesgue measurable.
 
  • #6
Samy_A
Science Advisor
Homework Helper
1,241
510
I feel there must be an easier one, but this should work.
The derivative of a function must be Lebesgue measurable.

Proof: assume ##f=g'##
That means that ##\displaystyle \forall x \in \mathbb R: \lim_{h \rightarrow 0} \frac{g(x+h)-g(x)}{h}=g'(x)=f(x)##.
Define ##f_n:\mathbb{R}\rightarrow \mathbb{R}: x \mapsto \frac{g(x+1/n)-g(x)}{1/n}##
Then ##\displaystyle \forall x \in \mathbb R: \lim_{n \rightarrow +\infty}f_n(x)=g'(x)=f(x)##.
Since ##g## is differentiable, it is also Lebesgue measurable. Hence the functions ##f_n## are also Lebesgue measurable.
##f## being the pointwise limit of Lebesgue measurable function is itself Lebesgue measurable.

So a counterexample is a function ##f## that is not Lebesgue measurable.
An easier one:

Define ##f:\mathbb{R}\rightarrow \mathbb{R}## as follows
##\left\{\begin{array}{l}
x \in \mathbb Q : x \mapsto 1\\
x \notin \mathbb Q : x \mapsto 0
\end{array}\right.##

Notice ##f=0## almost everywhere.

If ##f=g'##, then ##\displaystyle \int_0^a g'(x)dx=g(a)-g(0)##. (EDIT: I wonder: for this I probably need to first prove that g' is measurable, so we need the proof in the previous post anyway.)
But ##\displaystyle \int_0^a g'(x)dx =\int_0^a f(x)dx =0##.
Hence ##g## is constant, and its derivative must be 0. We have reached a contradiction.
 
Last edited:
  • #7
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
12,376
5,155
Number 4 seems quite easy. ##f(x) = sin(x^2)##

Let ##x_n = (n\pi)^{1/2}## and ##y_n = [(n + 1/2)\pi]^{1/2}##

##\forall n \ \ f(x_n) = sin(n\pi) = 0## and ## f(y_n) = sin(n\pi + \pi/2) = \pm 1##

hence ##|f(x_n) - f(y_n)| = 1##

But ##\lim_{n \rightarrow \infty}|x_n - y_n| = 0## (*)

Therefore, ##\forall \delta > 0## we can choose ##n## such that ##|x_n - y_n| < \delta## but ##|f(x_n) - f(y_n)| = 1##

Hence ##f## is not uniformly continuous.

(*) ##|x_n - y_n|(x_n + y_n) = \pi/2##

##|x_n - y_n| = \frac{\pi}{2(x_n + y_n)} < \frac{\pi}{2\sqrt{n\pi}}##
 
  • #8
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
12,376
5,155
I feel there must be an easier one, but this should work.
The derivative of a function must be Lebesgue measurable.

Proof: assume ##f=g'##
That means that ##\displaystyle \forall x \in \mathbb R: \lim_{h \rightarrow 0} \frac{g(x+h)-g(x)}{h}=g'(x)=f(x)##.
Define ##f_n:\mathbb{R}\rightarrow \mathbb{R}: x \mapsto \frac{g(x+1/n)-g(x)}{1/n}##
Then ##\displaystyle \forall x \in \mathbb R: \lim_{n \rightarrow +\infty}f_n(x)=g'(x)=f(x)##.
Since ##g## is differentiable, it is also Lebesgue measurable. Hence the functions ##f_n## are also Lebesgue measurable.
##f## being the pointwise limit of Lebesgue measurable function is itself Lebesgue measurable.

So a counterexample is a function ##f## that is not Lebesgue measurable.
Does that make this equivalent to the axiom of choice?
 
  • #9
Samy_A
Science Advisor
Homework Helper
1,241
510
Does that make this equivalent to the axiom of choice?
Think so (for the existence of not Lebesgue measurable functions). But you don't need the axiom of choice to prove that a derivative is Lebesgue measurable. So my second example probably works without invoking the axiom of choice.
 
  • #10
22,097
3,279
I feel there must be an easier one, but this should work.
The derivative of a function must be Lebesgue measurable.

Proof: assume ##f=g'##
That means that ##\displaystyle \forall x \in \mathbb R: \lim_{h \rightarrow 0} \frac{g(x+h)-g(x)}{h}=g'(x)=f(x)##.
Define ##f_n:\mathbb{R}\rightarrow \mathbb{R}: x \mapsto \frac{g(x+1/n)-g(x)}{1/n}##
Then ##\displaystyle \forall x \in \mathbb R: \lim_{n \rightarrow +\infty}f_n(x)=g'(x)=f(x)##.
Since ##g## is differentiable, it is also Lebesgue measurable. Hence the functions ##f_n## are also Lebesgue measurable.
##f## being the pointwise limit of Lebesgue measurable function is itself Lebesgue measurable.

So a counterexample is a function ##f## that is not Lebesgue measurable.
Here is another proof: https://en.wikipedia.org/wiki/Darboux's_theorem_(analysis)
 
  • #11
22,097
3,279
If ##f=g'##, then ##\displaystyle \int_0^a g'(x)dx=g(a)-g(0)##. (EDIT: I wonder: for this I probably need to first prove that g' is measurable, so we need the proof in the previous post anyway.)
This is a very interesting remark, and highly depends on what kind of integral you're looking at. Let's first look at the classical fundamental theorem of calculus (quoted from wiki):

wikipedia said:
Let ##f## and ##F## be real-valued functions defined on a closed interval ##[a,b]## such that ##F## is continuous and the derivative of ##F## is ##f##. That is, ##f## and ##F## are functions such that for all ##x## in ##(a, b)##,
[tex]F^\prime (x) = f(x)[/tex]
If ##f## is Riemann integrable on ##[a,b]## then
[tex]\int_a^b f(x)dx = F(b) - F(a)[/tex]
So we see in your counterexample, that you will first need to prove that ##g^\prime(x)## is Riemann integrable. This is obviously not the case. The classical fundamental theorem is not strong enough to let you do what you want.

Luckily, there is an extension of the Riemann integral, called the Henstock integral. This integral has a very similar definition to the Riemann integral, but has an improved fundamental theorem. Its theory is explained in Bartle "A modern theory of integration" https://www.amazon.com/dp/0821808451/?tag=pfamazon01-20 Let me quote the relevant part

Bartle said:
If ##f : [a, b] \rightarrow \mathbb{R}## has a primitive ##F## on ##[a,b]##, then ##f## is Henstock integrable and
[tex]\int_a^b f(x)dx = F(b) - F(a)[/tex]
(So this fundamental theorem actually proves Henstock integrability of ##f##. No further assumptions are necessary other than it being a derivative. Not even measurability. (Note that Henstock integrability implies measurable though). So this theorem does exactly what you want. Note that this isn't even the most general theorem of this kind, you can get away with a lot less than what the quoted part assumed, all of that is in Bartle.

Now I could be very justified to put in a rant on why Henstock integrals are not at all taught in schools. They are extremely easy to define, and are more general than Lebesgue integrals (for functions ##\mathbb{R}\rightarrow \mathbb{R}##), and they have way better properties than the silly Riemann integral. Sadly, they are ignored by the vast majority of books (one notable exception is DePree Swartz https://www.amazon.com/dp/0471853917/?tag=pfamazon01-20)
 
  • #12
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
12,376
5,155
I assume for number 1 the statement should be "not" and then the counterexample is R given as a disjoint countable union of closed intervals. I'll do non-negative R.

The first set of intervals are ##[0, 1], [2, 3], [4, 5] \dots##

The second set of intervals are ##[1\frac{1}{4}, 1\frac{3}{4}], [3\frac{1}{4}, 3\frac{3}{4}] \dots##

The third set of intervals are ##[1\frac{1}{16}, 1\frac{3}{16}], [1\frac{13}{16}, 1\frac{15}{16}], [3\frac{1}{16}, 3\frac{3}{16}], [3\frac{13}{16}, 1\frac{15}{16}] \dots##

Etc.
 
  • #13
jbriggs444
Science Advisor
Homework Helper
2019 Award
8,322
3,154
I assume for number 1 the statement should be "not" and then the counterexample is R given as a disjoint countable union of closed intervals. I'll do non-negative R.
But.

The only rub now is that only 9 are false, thus one of the statements is true. Provide a counterexample to the false statements and a proof for the correct one!
It seems that you have provided at least the core of a proof for the correct one.
 
  • #14
jbriggs444
Science Advisor
Homework Helper
2019 Award
8,322
3,154
For any infinitely differentiable monotonic function ##f:\mathbb{R}\rightarrow \mathbb{R}## such that ##\lim_{x\rightarrow +\infty} f(x) = 0##, also holds that ##\lim_{x\rightarrow +\infty} f^\prime(x) = 0##.
The idea is that one wants to come up with a kind of smoothed step function where the steps get smaller and smaller but retain a similar slope. If one had as a building block a monotone increasing function that is continuous and infinitely differentiable on the unit interval [0,1] with all derivatives equal to zero at both endpoints then one could use that to construct the counter example by splining such building blocks together. Unfortunately, my toolbag only contains a function with the required properties at one endpoint.

Wikipedia calls these "smooth transfer functions". And I think at this point I think I've been spoiled too badly to compete.
 
  • #15
22,097
3,279
I assume for number 1 the statement should be "not" and then the counterexample is R given as a disjoint countable union of closed intervals. I'll do non-negative R.

The first set of intervals are ##[0, 1], [2, 3], [4, 5] \dots##

The second set of intervals are ##[1\frac{1}{4}, 1\frac{3}{4}], [3\frac{1}{4}, 3\frac{3}{4}] \dots##

The third set of intervals are ##[1\frac{1}{16}, 1\frac{3}{16}], [1\frac{13}{16}, 1\frac{15}{16}], [3\frac{1}{16}, 3\frac{3}{16}], [3\frac{13}{16}, 1\frac{15}{16}] \dots##

Etc.
So, the easiest is to write this in base ##4## notation. Let me focus on the cover of ##(1,2)## which according to you is:

[tex][0.1, 0.3][/tex]
[tex]0.01, 0.03],~[0.31, 0.33][/tex]
[tex][0.001, 0.003],~[0.031, 0.033],~[0.0301, 0.303],~[0.331, 0.333][/tex]
And so on.

So the first step covers points of the form ##0.1xxxx...## and ##0.2xxx...## (assume terminating "decimals" are forbidden). No point with ##1## and ##2## in the first place remains in the remainder.
The second step covers points of the form ##0.01xxx...## and ##0.02xxx...## and ##0.31xxx...## and ##0.32xxx##. Removing this, we see that no point with ##1## and ##2## in the first two places remain.

All in all, in the ##n##th step, we see that no point with ##1## and ##2## in the first ##n## decimals places remain in the remainder.

So what your collection of closed intervals covers are all points with ##1## and ##2## in some decimal place. But from this description it is easy to see that the point ##0.030303030303...## is not covered by your collection of intervals. So your counterexample is not valid.
 
  • #16
22,097
3,279
And I think at this point I think I've been spoiled too badly to compete.
Don't worry, you can still compete with this question if you want to. After all, checking wikipedia or textbooks for specific subjects is allowed, as long as you don't find a direct answer to the question.
 
  • #17
Samy_A
Science Advisor
Homework Helper
1,241
510
Aahh, that's the one!
I knew there was some well known property that all derivatives had, but couldn't remember which one.
This is a very interesting remark, and highly depends on what kind of integral you're looking at. Let's first look at the classical fundamental theorem of calculus (quoted from wiki):



So we see in your counterexample, that you will first need to prove that ##g^\prime(x)## is Riemann integrable. This is obviously not the case. The classical fundamental theorem is not strong enough to let you do what you want.

Luckily, there is an extension of the Riemann integral, called the Henstock integral. This integral has a very similar definition to the Riemann integral, but has an improved fundamental theorem. Its theory is explained in Bartle "A modern theory of integration" https://www.amazon.com/dp/0821808451/?tag=pfamazon01-20 Let me quote the relevant part



(So this fundamental theorem actually proves Henstock integrability of ##f##. No further assumptions are necessary other than it being a derivative. Not even measurability. (Note that Henstock integrability implies measurable though). So this theorem does exactly what you want. Note that this isn't even the most general theorem of this kind, you can get away with a lot less than what the quoted part assumed, all of that is in Bartle.

Now I could be very justified to put in a rant on why Henstock integrals are not at all taught in schools. They are extremely easy to define, and are more general than Lebesgue integrals (for functions ##\mathbb{R}\rightarrow \mathbb{R}##), and they have way better properties than the silly Riemann integral. Sadly, they are ignored by the vast majority of books (one notable exception is DePree Swartz https://www.amazon.com/dp/0471853917/?tag=pfamazon01-20)
Very cool. Henstock integral: something to look at.
 
  • #18
22,097
3,279
Aahh, that's the one!
I knew there was some well known property that all derivatives had, but couldn't remember which one.
Well, we have detailed some of its properties already: derivatives are Lebesgue measurable and satisfy the conclusion of the intermediate value theorem. One could ask whether this completely characterizes derivatives. But that's for a next set of counterexamples (the answer is no).
 
  • #19
12,676
9,200
6. If ##f:\mathbb{R}\rightarrow \mathbb{R}## is continuous and sends open sets to open sets, then it also sends closed sets to closed sets.
I define ##f(x) = x^{-2}## for ##x\neq 0## and ##f(0)=0##.
My "fraud (?)" is this: I take the discrete topology on both ##\mathbb{R}##. Then ##f## is clearly open.
With ##f^{-1}(\{x\}) = \{\frac{1}{\sqrt{x}}\}## for ##x > 0##, ##f^{-1}(\{0\})=\{0\}## and ##f^{-1}(\{x\})=∅## for ##x < 0## ##f## is also continuous.
Now ##f(\mathbb{R} - \{0\}) = \mathbb{R^+} = ∪_{x>0} \{x\}## maps a closed set onto an open.
 
Last edited:
  • #20
22,097
3,279
I define ##f(x) = x^{-2}## for ##x\neq 0## and ##f(0)=0##.
My "fraud (?)" is this: I take the discrete topology on both ##\mathbb{R}##. Then ##f## is clearly open.
With ##f^{-1}(\{x\}) = \{\frac{1}{\sqrt{x}}\}## for ##x > 0##, ##f^{-1}(\{0\})=\{0\}## and ##f^{-1}(\{x\})=∅## for ##x < 0## ##f## is also continuous.
Now ##f(\mathbb{R} - \{0\}) = \mathbb{R^+} = ∪_{x>0} \{x\}## maps a closed set onto an open.
OK, but your map still takes closed sets to closed sets.

(And I kind of meant ##\mathbb{R}## to have the Euclidean topology)
 
  • #21
12,676
9,200
OK, but your map still takes closed sets to closed sets.

(And I kind of meant ##\mathbb{R}## to have the Euclidean topology)
So you want to have all closed sets not to be closed? Even ##f(\mathbb{R})##?
 
  • #22
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
12,376
5,155
It seems to me that for number 6, you have to exclude ##\mathbb{R}## itself as an allowable closed set. Any monotonic bounded function such as ##arctan## would map ##\mathbb{R}## to an open interval. That seems a trivial counterexample otherwise.
 
  • #23
22,097
3,279
So you want to have all closed sets not to be closed? Even ##f(\mathbb{R})##?
No, I just want one image of a closed set that is not closed.
 
  • #24
22,097
3,279
It seems to me that for number 6, you have to exclude ##\mathbb{R}## itself as an allowable closed set. Any monotonic bounded function such as ##arctan## would map ##\mathbb{R}## to an open interval. That seems a trivial counterexample otherwise.
Yes, that's a right solution. Not all of them need to be hard :woot:
 
  • #25
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
12,376
5,155
Hopefully proof for number 1: all sequences of closed intervals are essentially the same. Focus on ##(0, 1)## and show that can't be covered:

Let's assume we have a valid sequence of closed disjoint intervals covering ##(0,1)##. The first interval is ##I_1 = (x_1, y_1)## and the set remaining is:

##R_1 = (0, x_1) \cup (y_1, 1)##

Assume, wlog, that the second interval is a subset of the first interval: ##I_2 = (x_2, y_2) \subset (0, x_1)##

##R_2 = (0, x_2) \cup (y_2, x_1) \cup (y_1, 1)##

We're actually going to take a subsequence of the original sequence, so to save on subscripts I'll take ##I_3 = (x_3, y_3) \subset (y_2, x_1)## (the next interval that splits this remaining interval):

##R_3 \supset (y_2, x_3) \cup (y_3, x_1)##

And, take ##I_4 = (x_4, y_4) \subset (y_2, x_3)##

##R_4 \supset (y_2, x_4) \cup (y_4, x_3) ##

And, take ##I_5 = (x_5, y_5) \subset (y_4, x_3)##

##R_5 \supset (y_4, x_5) \cup (y_5, x_3)##

Continuing in this way, we can generate a sequence of open subsets:

##(0, x_1) \supset (y_2, x_3) \supset (y_4, x_5) \dots##

And the intersection of the ##R_n## sets contains the intersection of these sets.

Finally, by the completeness of ##\mathbb{R}## there must be a point in this intersection. We can generate a monotone sequence from the ##y_n## endpoints that must converge to some ##y##. If ##y## is not in the intersection, then it must be outside one of the sets. But, the next set has no endpoints in common with the previous one, so it must be a finite distance outside that and subsequent sets, which is a contradiction.

This non-empty intersection is outside the union of all the ##I_n##, which is a contradiction to the original assumption.

(Note that the same argument can apply to every interval generated, so there must be (at least) a countable number of points outside the union.)
 

Related Threads on Yet another counterexample challenge!

Replies
62
Views
5K
Replies
55
Views
3K
Replies
105
Views
8K
Replies
116
Views
11K
Replies
35
Views
3K
Replies
77
Views
7K
  • Last Post
6
Replies
125
Views
11K
Replies
74
Views
6K
Replies
42
Views
4K
  • Last Post
Replies
3
Views
591
Top