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239 votes
14 answers
76k views

Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...
238 votes
10 answers
43k views

If $f$ is infinitely differentiable then $f$ coincides with a polynomial

Let $f$ be an infinitely differentiable function on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then does $f$ coincide on $[0,1]$ ...
C.S.'s user avatar
  • 4,795
155 votes
4 answers
18k views

Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
Willie Wong's user avatar
141 votes
17 answers
38k views

Why is differentiating mechanics and integration art?

It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by parts/...
vonjd's user avatar
  • 5,935
122 votes
5 answers
27k views

Is the series $\sum_n|\sin n|^n/n$ convergent?

Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent? (The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
Lviv Scottish Book's user avatar
114 votes
34 answers
86k views

Why do we teach calculus students the derivative as a limit?

I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students? Something a teacher ...
107 votes
9 answers
36k views

solving $f(f(x))=g(x)$

This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
Kevin Buzzard's user avatar
106 votes
5 answers
10k views

integral of a "sin-omial" coefficients=binomial

I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof? For any pair of integers $n\geq k\geq0$, we have $$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
T. Amdeberhan's user avatar
102 votes
21 answers
15k views

Proofs of the uncountability of the reals

Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem (Wayback Machine). At first, I was excited to see a variant proof (as it did not ...
Unknown's user avatar
  • 2,855
89 votes
7 answers
13k views

If I exchange infinitely many digits of $\pi$ and $e$, are the two resulting numbers transcendental?

If I swap the digits of $\pi$ and $e$ in infinitely many places, I get two new numbers. Are these two numbers transcendental?
user avatar
89 votes
1 answer
21k views

Is the largest root of a random polynomial more likely to be real than complex?

This question might be hard because it got $35$ upvotes in MSE and also had a $200$ points bounty by Jyrki Lahtonen but it was unanswered. So I am posting it in MO. The number of real roots of a ...
Nilotpal Kanti Sinha's user avatar
87 votes
8 answers
16k views

Why is Lebesgue integration taught using positive and negative parts of functions?

Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only ...
KConrad's user avatar
  • 50.6k
80 votes
4 answers
9k views

Who first characterized the real numbers as the unique complete ordered field?

Nearly every mathematician nowadays is familiar with the fact that there is up to isomorphism only one complete ordered field, the real numbers. Theorem. Any two complete ordered fields are isomorphic....
Joel David Hamkins's user avatar
78 votes
5 answers
8k views

Does pointwise convergence imply uniform convergence on a large subset?

Suppose $f_n$ is a sequence of real valued functions on $[0,1]$ which converges pointwise to zero. Is there an uncountable subset $A$ of $[0,1]$ so that $f_n$ converges uniformly on $A$? Is there a ...
Bill Johnson's user avatar
  • 31.5k
74 votes
15 answers
18k views

$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential

The question is about the function $f(x)$ so that $f(f(x))=\exp (x)-1$. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://...
Gil Kalai's user avatar
  • 24.7k
72 votes
9 answers
16k views

Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)

Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
Yoo's user avatar
  • 1,093
68 votes
2 answers
2k views

Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...
Abcd's user avatar
  • 629
67 votes
9 answers
7k views

Taking "Zooming in on a point of a graph" seriously

In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation ...
Steven Gubkin's user avatar
64 votes
8 answers
6k views

Two (probably) equal real numbers which are not proved to be equal?

Can someone give me a nice example of two computable real numbers which are believed but not proved to be equal? I never really understood the assertion that "the reals do not have decidable equality"...
63 votes
6 answers
12k views

Why isn't integral defined as the area under the graph of function?

In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also ...
user57888's user avatar
  • 1,229
61 votes
1 answer
5k views

Every real function has a dense set on which its restriction is continuous

The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous. Or so I'm told, but this leaves me ...
Gro-Tsen's user avatar
  • 32.5k
59 votes
1 answer
5k views

Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.
DLN's user avatar
  • 817
54 votes
3 answers
4k views

Does every real function have this weak continuity property?

In my research I came across the following question : Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x_n)_n$, taking infinitely many values, ...
Dattier's user avatar
  • 4,074
54 votes
3 answers
6k views

On which regions can Green's theorem not be applied?

In elementary calculus texts, Green's theorem is proved for regions enclosed by piecewise smooth, simple closed curves (and by extension, finite unions of such regions), including regions that are not ...
GermanJablo's user avatar
54 votes
4 answers
3k views

When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true. For example, it gives some evidence that there are finitely many ...
Eric Naslund's user avatar
  • 11.4k
54 votes
4 answers
12k views

Everywhere differentiable function that is nowhere monotonic

It is well known that there are functions $f \colon \mathbb{R} \to \mathbb{R}$ that are everywhere continuous but nowhere monotonic (i.e. the restriction of $f$ to any non-trivial interval $[a,b]$ is ...
Ricky's user avatar
  • 3,704
52 votes
7 answers
6k views

On an example of an eventually oscillating function

For $x\in(0,1)$, put $$f(x):=\sum_{n=0}^{\infty}(-1)^{n}x^{2^{n}}.$$ This function possesses interesting properties. It grows monotonically from $0$ up to certain point. Then it starts to oscillate ...
Twi's user avatar
  • 2,188
51 votes
4 answers
17k views

Function satisfying $f^{-1} =f'$

How many functions are there which are differentiable on $(0,\infty)$ and that satisfy the relation $f^{-1}=f'$?
C.S.'s user avatar
  • 4,795
51 votes
5 answers
18k views

Integrability of derivatives

Is there a (preferably simple) example of a function $f:(a,b)\to \mathbb{R}$ which is everywhere differentiable, such that $f'$ is not Riemann integrable? I ask for pedagogical reasons. Results in ...
Mark Meckes's user avatar
  • 11.4k
51 votes
6 answers
2k views

Why does $d^n \exp(-x-x^{-1})/(dx)^n$ only have $n$ positive real zeroes?

Set $f(x) = \exp(-x-x^{-1})$. An easy induction shows that $$\frac{d^n}{(dx)^n} f(x) = \phi_n(x^{-1}) f(x)$$ for $\phi_n$ a polynomial of degree $2n$. Clearly, the roots of $\phi_n(x^{-1})$ are the ...
David E Speyer's user avatar
49 votes
2 answers
3k views

Is a function with nowhere vanishing derivatives analytic?

My question is the following: Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic? EDIT. According to a ...
smyrlis's user avatar
  • 2,933
49 votes
3 answers
6k views

The Hardy Z-function and failure of the Riemann hypothesis

David Feldman asked whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly ...
David Hansen's user avatar
  • 13.1k
46 votes
7 answers
10k views

Are some numbers more irrational than others?

Some irrational numbers are transcendental, which makes them in some sense "more irrational" than algebraic numbers. There are also numbers, such as the golden ratio $\varphi$, which are poorly ...
I. J. Kennedy's user avatar
46 votes
4 answers
8k views

Why could Mertens not prove the prime number theorem?

We know that $$ \sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x) $$ where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy $$ \sum_{p \le x}\frac{1}{p} = \ln\ln ...
Nilotpal Kanti Sinha's user avatar
46 votes
2 answers
6k views

Is the following identity true?

Calculation suggests the following identity: $$ \lim_{n\to \infty}\sum_{k=1}^{n}\frac{(-1)^k}{k}\sum_{j=1}^k\frac{1}{2j-1}=\frac{1-\sqrt{5}}{2}. $$ I have verified this identity for $n$ up to $5000$ ...
Chitsai Liu's user avatar
  • 2,183
46 votes
2 answers
7k views

Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?

I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
npbool's user avatar
  • 573
46 votes
2 answers
8k views

"Closed-form" functions with half-exponential growth

Let's call a function f:N→N half-exponential if there exist constants 1<c<d such that for all sufficiently large n, cn < f(f(n)) < dn. Then my question is this: can we prove that no ...
Scott Aaronson's user avatar
45 votes
5 answers
3k views

An "analytic continuation" of power series coefficients

Cauchy residue theorem tells us that for a function $$f(z) = \sum_{k \in \mathbb{Z}} a(k) z^k,$$ the coefficient $a(k)$ can be extracted by an integral formula $$a(k) = \frac{1}{2\pi i}\oint f(z) z^{-...
MCH's user avatar
  • 1,324
44 votes
10 answers
47k views

Is square of Delta function defined somewhere?

I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere. In the beginning, this question might look strange. But by restricting the space of the test functions, ...
44 votes
7 answers
4k views

The missing link: an inequality

I've been working on a project and proved a few relevant results, but got stuck on one tricky problem: Conjecture. If $2\leq n\in\mathbb{N}$ and $0<x<1$ is a real number, then $$F_n(x)=\...
T. Amdeberhan's user avatar
44 votes
3 answers
4k views

Smooth functions for which $f(x)$ is rational if and only if $x$ is rational

A friend of mine introduced me to the following question: Does there exist a smooth function $f: \mathbb{R} \to \mathbb{R}$, ($f \in C^\infty$), such that $f$ maps rationals to rationals and ...
J. J.'s user avatar
  • 543
43 votes
2 answers
4k views

Square root of a positive $C^\infty$ function.

Suppose $f$ is a $C^\infty$ function from the reals to the reals that is never negative. Does it have a $C^\infty$ square root? Clearly the only problem points are those at which $f$ vanishes.
Michael  Barr's user avatar
43 votes
0 answers
819 views

A kaleidoscopic coloring of the plane

Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
Taras Banakh's user avatar
  • 41.8k
41 votes
6 answers
9k views

"Long-standing conjectures in analysis ... often turn out to be false"

The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1 His example of a "long-standing conjecture" is the Riemann hypothesis,...
40 votes
5 answers
10k views

Is there a natural measures on the space of measurable functions?

Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...
Kenny Easwaran's user avatar
39 votes
8 answers
13k views

Can Cantor set be the zero set of a continuous function?

More generally, can the zero set $V(f)$ of a continuous function $f : \mathbb{R} \to \mathbb{R}$ be nowhere dense and uncountable? What if $f$ is smooth? Some days ago I discovered that in this proof ...
pinaki's user avatar
  • 5,339
38 votes
26 answers
57k views

Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
38 votes
13 answers
5k views

Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful? I ...
Lehs's user avatar
  • 862
38 votes
4 answers
3k views

Binomial again, and again

Let $\lceil a\rceil=$ the smallest integer $\geq a$, otherwise known as the ceiling function. When the arguments are real, interpret $\binom{a}b$ using the Euler's gamma function, $\Gamma$. Recently, ...
T. Amdeberhan's user avatar
37 votes
12 answers
5k views

Examples where existence is harder than evaluation

In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, ...

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