Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may ...
175
votes
23answers
7k views
Examples of apparent patterns that eventually fail
Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
115
votes
28answers
8k views
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...
102
votes
18answers
8k views
Nice examples of groups which are not obviously groups
I am searching for some groups, where it is not so obvious that they are groups.
In the lectures script there are only examples like $\mathbb{Z}$ under
addition and other things like that. I ...
101
votes
23answers
5k views
Can't argue with success? Looking for “bad math” that “gets away with it”
I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare").
One example would be "cancelling" the 6s in
$$\frac{64}{16}.$$
...
41
votes
3answers
1k views
Is it possible for a function to be in $L^p$ for only one $p$?
I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain).
One can use interpolation to show that ...
39
votes
1answer
1k views
How discontinuous can a derivative be?
There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many ...
38
votes
11answers
1k views
Examples of results failing in higher dimensions
A number of economists do not appreciate rigor in their usage of mathematics and I find it very discouraging.
One of the examples of rigor-lacking approach are proofs done via graphs or pictures ...
29
votes
3answers
1k views
What is the spectral theorem for compact self-adjoint operators on a Hilbert space actually for?
Please excuse the naive question. I have had two classes now in which this theorem was taught and proven, but I have only ever seen a single (indirect?) application involving the quantum harmonic ...
26
votes
5answers
1k views
False beliefs about Lebesgue measure on $\mathbb{R}$
I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
25
votes
17answers
3k views
Examples of mathematical induction
What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
24
votes
5answers
1k views
An example of an easy to understand undecidable problem
I am looking for an undecidable problem that I could give as an easy example in a presentation to the general public. I mean easy in the sense that the mathematics behind it can be described, well, ...
23
votes
3answers
804 views
If every continuous $f:X\to X$ has $\text{Fix}(f)\subseteq X$ closed, must $X$ be Hausdorff?
Given a function $f:X\to X$, let $\text{Fix}(f)=\{x\in X\mid x=f(x)\}$.
In a recent comment, I wondered whether
$X$ is Hausdorff $\iff$ $\text{Fix}(f)\subseteq X$ is closed for every continuous ...
22
votes
2answers
255 views
Is every function with the intermediate value property a derivative?
As it is well known every continuous function has the intermediate value property, but even some discontinuous functions like
$$f(x)=\left\{
\begin{array}{cl}
\sin\left(\frac{1}{x}\right) & x\neq ...
20
votes
17answers
988 views
Accidents of small $n$
In studying mathematics, I sometimes come across examples of general facts that hold for all $n$ greater than some small number. One that comes to mind is the Abel–Ruffini theorem, which states that ...
20
votes
2answers
709 views
If a continuous function is positive on the rationals, is it positive almost everywhere?
I made up this question, but unable to solve it:
Let $f : \mathbb R \to \mathbb R$ be a continuous function such that $f(x) > 0$ for all $x \in \mathbb Q$. Is it necessary that $f(x) > 0$ ...
18
votes
16answers
2k views
An example for a calculation where imaginary numbers are used but don't occur in the question or the solution.
In a presentation I will have to give an account of Hilbert's concept of real and ideal mathematics. Hilbert wrote in his treatise "Über das Unendliche" (page 14, second paragraph. Here is an English ...
18
votes
2answers
312 views
What is a metric for $\mathbb Q$ in the lower limit topology?
A useful source of counterexamples in topology is $\mathbb R_\ell$, the set $\mathbb R$ together with the lower limit topology generated by half-open intervals of the form $[a,b)$. For example this ...
17
votes
6answers
809 views
Uncountable closed set of irrational numbers
Could you construct an actual example of a uncountable set of irrational numbers that is closed (in the topological sense)?
I can find countable examples that are closed, like $\{ \sqrt{2} + ...
17
votes
4answers
432 views
Is the closure of a Hausdorff space, Hausdorff?
$(X,\mathcal T)$ is a topological space which has a dense Hausdorff subspace. Is $X$ Hausdorff?
17
votes
3answers
2k views
Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?
I know that if $f$ is a Lebesgue measurable function on $[a,b]$ then there exists a continuous function $g$ such that $|f(x)-g(x)|< \epsilon$ for all $x\in [a,b]\setminus P$ where the measure of ...
17
votes
3answers
322 views
Why does the Hilbert curve fill the whole square?
I have never seen a formal definition of the Hilbert curve, much less a careful analysis of why it fills the whole square. The Wikipedia and Mathworld articles are typically handwavy.
I suppose the ...
17
votes
2answers
341 views
Basic counterexample re: preimages of ideals
I'm trying to think of an example of a homomorphism of commutative rings $f:A\rightarrow B$ and ideals $I,J$ of $B$ such that $f^{-1}(I)+f^{-1}(J)$ is not a preimage of any ideal of $B$. I can't seem ...
16
votes
1answer
1k views
An example of a division ring $D$ that is **not** isomorphic to its opposite ring
I recall reading in an abstract algebra text two years ago (when I had the pleasure to learn this beautiful subject) that there exists a division ring $D$ that is not isomorphic to its opposite ring. ...
16
votes
3answers
669 views
Examples of math contest problems that can be solved in a 'cheap' way?
What are some examples of math contest problems that can be solved by using a nonrigorous, 'cheap' shortcut?
For instance, a problem on the 2011 AMC went:
A raft and a motorboat left dock A and ...
16
votes
1answer
1k views
Isomorphic quotients by isomorphic normal subgroups
In this recent question, Iota asked if, given a finite group $G$ and two isomorphic normal subgroups $H$ and $K$, it would follow that $G/H$ and $G/K$ are isomorphic. This is not true (a simple ...
15
votes
5answers
566 views
Can two topological spaces surject onto each other but not be homeomorphic?
Let $X$ and $Y$ be topological spaces and $f:X\rightarrow Y$ and $g:Y\rightarrow X$ be surjective continuous maps. Is it necessarily true that $X$ and $Y$ are homeomorphic? I feel like the answer to ...
15
votes
4answers
559 views
Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?
Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?
I think it is probable because we can consider
$$ y ...
15
votes
2answers
783 views
examples of measurable functions on $\mathbb{R}$
Could give some examples of nonnegative measurable function $f:\mathbb{R}\to[0,\infty)$, such that its integral over any bounded interval is infinite?
15
votes
2answers
807 views
How useless can the Mayer-Vietoris sequence be in general?
In an algebraic topology course I'm taking we are often asked to compute the homology groups of a space $X = A \cup B$ using the Mayer-Vietoris sequence, and it happens in all of the examples I've ...
15
votes
2answers
553 views
Clarifying the relationship between outer measures, measures and measurable spaces: the converse direction
This is related to my measure theory class, but it's not homework. The motivation behind this post is to understand the exact relationship between the three objects mentioned in the title.
...
14
votes
5answers
204 views
Looking for Cover's hubris-busting ${\mathbb R}^{N\gg3}$ counterexamples
In lecture 4 of his Introduction to Dynamical Linear Systems course, right after interpreting the inner product in ${\mathbb R}^N$ in terms of the cosine of the subtended angle, Stanford's Stephen ...
14
votes
3answers
395 views
Weird subfields of $\Bbb{R}$
I found this problem, and I can't get an answer to it:
Prove that there are subfields of $\Bbb{R}$ that are
a) non-measurable.
b) of measure zero and continuum cardinality.
I can't ...
14
votes
2answers
212 views
Construct an example of set $A$ for which $A+A=A $ but $0∉cl(A)$
How to prove that convexity is necessary condition in this question? Need to construct an example of set $A$ for which $A+A=A$ but $0 \notin cl(A)$. From the linked question follows that $A$ must be ...
14
votes
2answers
281 views
Is every group the automorphism group of a group?
Suppose $G$ is a group. Does there always exist a group $H$, such that $\operatorname{Aut}(H)=G$, i. e. such that $G$ is the automorphism group of $H$?
EDIT: It has been pointed out that the answer ...
14
votes
1answer
534 views
A counterexample in topology
Semi-local simple connectedness is a property that arises in Algebraic Topology in the study of covering spaces, namely, it is a necessary condition for the existence of the universal cover of a ...
13
votes
9answers
522 views
Examples of nonabelian groups.
Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
13
votes
2answers
625 views
“Pseudo-Cauchy” sequences: are they also Cauchy?
I tried to prove something but I could not, I don't know if it's true or not, but I did not found a counterexample.
Let $(a_n)$ be a sequence in a general metric space such that for any fixed $k ...
13
votes
4answers
569 views
Is a vector space over a finite field always finite?
Definition of a vector space:
Let $V$ be a set and $(\mathbb{K}, +, \cdot)$ a field.
$V$ is called a vector space over the field $\mathbb{K}$ if:
V1: $(V, +)$ is a commutative group
V2: $\forall ...
13
votes
3answers
1k views
Discontinuous linear functional
I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, ...
13
votes
4answers
437 views
Example of a rational function such that : $(f(x))^{3} + (g(x))^{3} + (h(x))^{3}=x$
Can any one give me example of: rational functions $f, g$ and $h$ with rational coefficients such that
$$(f(x))^{3} + (g(x))^{3} + (h(x))^{3}=x$$
Also, if anyone knows a procedure for constructing ...
13
votes
1answer
210 views
Measurable subset of $\mathbb{R}$ with a specific property
Let $A$ be a subset of $\mathbb{R}$ such that its intersection with every finite segment is Lebesgue measurable. I am looking for an example of such an $A$ with the additional property that the ...
12
votes
2answers
374 views
Why is $T_1$ required for a topological space to be $T_4$?
Let's say we have some topological space.
Axiom $T_1$ states that for any two points $y \neq x$, there is an open neighborhood $U_y$ of $y$ such that $x \notin U_y$.
Then we say that a topological ...
12
votes
3answers
383 views
Why does the Continuum Hypothesis make an ideal measure on $\mathbb R$ impossible?
On the page 43 of Real Analysis by H.L. Royden (1st Edition) we read: "(Ideally) we should like $m$ (the measure function) to have the following properties:
$m(E)$ is defined for each subset $E$ of ...
12
votes
1answer
316 views
Complete space as a disjoint countable union of closed sets
It is a consequence of Baire's theorem that a connected, locally connected complete space cannot be written
$$ X = \bigcup_{n \geq 1}\ F_n$$
where the $F_n$ are nonempty, pairwise disjoint closed ...
12
votes
1answer
192 views
Examples of universal constructions in probability theory
I am looking for examples of universal constructions in probability theory. Like the construction a of Gaussian space from a real Hilbert space, or a Poisson jump process from a measurable space with ...
11
votes
2answers
844 views
Comparing the Lebesgue measure of an open set and its closure
Let $E$ be an open set in $[0,1]^n$ and $m$ be the Lebesgue measure.
Is it possible that $m(E)\neq m(\bar{E})$, where $\bar{E}$ stands for the closure of $E$?
11
votes
4answers
271 views
Counterexamples in complex analysis
In contrast to other topics in analysis such as functional analysis with its vast amount of counterexamples to intuitively correct looking statements (see here for an example), everything in complex ...
11
votes
2answers
360 views
A (non-artificial) example of a ring without maximal ideals
As a brief overview of the below, I am asking for:
An example of a ring with no maximal ideals that is not a zero ring.
A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ...
11
votes
3answers
271 views
Nasty examples for different classes of functions
Let $f: \mathbb{R} \to \mathbb{R}$ be a function. Usually when proving a theorem where $f$ is assumed to be continuous, differentiable, $C^1$ or smooth, it is enough to draw intuition by assuming ...
11
votes
2answers
523 views
Set of zeroes of the derivative of a pathological function
For a continuous function $f : [0,1] \to {\mathbb R}$, let us set
$$
X_f=\lbrace x \in [0,1] \bigg| f'(x)=0 \rbrace
$$
(for a $x\not\in X_f$, $f'(x)$ may be a nonzero value or undefined).
There ...
