**139**

votes

**11**answers

42k 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 ...

**112**

votes

**8**answers

12k 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 $[0,1]$ ...

**83**

votes

**17**answers

19k 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 ...

**67**

votes

**9**answers

20k 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 ...

**62**

votes

**32**answers

42k 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 ...

**57**

votes

**5**answers

4k 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 ...

**55**

votes

**18**answers

7k 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. At first, I was excited to see a variant proof (as it did not use the diagonal ...

**52**

votes

**9**answers

3k 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 ...

**52**

votes

**8**answers

7k 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 ...

**50**

votes

**1**answer

642 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 ...

**48**

votes

**4**answers

2k 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 ...

**47**

votes

**16**answers

10k 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 ...

**43**

votes

**2**answers

2k 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 ...

**37**

votes

**3**answers

1k 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 ...

**34**

votes

**6**answers

2k 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 ...

**34**

votes

**2**answers

4k 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 ...

**34**

votes

**4**answers

2k 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 ...

**33**

votes

**7**answers

5k 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 ...

**31**

votes

**19**answers

6k views

### Interesting applications (in pure mathematics) of first-year calculus

What interesting applications are there for theorems or other results studied in first-year calculus courses?
A good example for such an application would be using a calculus theorem to prove a ...

**31**

votes

**1**answer

1k views

### Is $π$ definable in $(\Bbb R,0,1,+,×,<,\exp)$?

(This question is originally from Math.SE, where it didn't receive any answers.)
Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields ...

**31**

votes

**1**answer

771 views

### For which maps $S^1\to S^1$ is the winding number defined?

There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:
• Continuous maps:
Using the unique path lifting property of the universal covering map $\mathbb R\to ...

**31**

votes

**1**answer

1k views

### Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}\setminus (-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least $n+1$ zeros on $(-1,1)$
...

**30**

votes

**2**answers

1k 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.

**30**

votes

**1**answer

1k 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 ...

**30**

votes

**3**answers

3k 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 ...

**30**

votes

**1**answer

2k views

### What did Rolle prove when he proved Rolle's theorem?

Rolle published what we today call Rolle's theorem about 150 years before the arithmetization of the reals. Unfortunately this proof seems to have been buried in a long book [Rolle 1691] that I can't ...

**30**

votes

**1**answer

2k views

### An Entropy Inequality

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...

**28**

votes

**2**answers

2k 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 ...

**27**

votes

**2**answers

1k views

### Continuous functions $f$ with $f(A)$ linearly independent when $A$ is independent

Is there any characterization of continuous functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that for any linearly independent set $A$ (over the rationals) $f(A)$ is also linearly independent ?

**27**

votes

**5**answers

3k views

### Differentiable functions with discontinuous derivatives

For years I've taught my honors calculus students about functions like (the continuous extension of) $x^2 \sin 1/x$, and for just as many years I've told them that they won't encounter functions like ...

**26**

votes

**9**answers

17k views

### Is square of Delta function defined somewhere?

Hello, every one. I am wondering whether any one knows that whether the square of Dirac Delta function is defined some where?
In the beginning, this question might look strange. But by restricting ...

**25**

votes

**2**answers

1k 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 $\sum_{k=1}^{n} \sin(k^2)$?

**25**

votes

**9**answers

2k views

### Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful?
I have a vague idea of the possibility of ...

**25**

votes

**1**answer

1k views

### Is pi = log_a(b) for some integers a, b > 1?

Are there integers $a, b > 1$ such that $\pi = \log_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - ...

**24**

votes

**4**answers

6k 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 ...

**23**

votes

**4**answers

1k views

### is f a polynomial provided that it is “partially” smooth?

Let $f$ be a $C^\infty$ function on $(c,d)$ ,and
let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$.
Suppose for each $n\in ...

**23**

votes

**2**answers

756 views

### Codimension of Measurable Sets

I am currently teaching an advanced undergraduate analysis class, and the following question came up.
Intuition suggests that "most" subsets of $[0,1]$ are not Lebesgue measurable. However, the ...

**22**

votes

**8**answers

5k 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?
About two days ago I discovered that in this ...

**22**

votes

**7**answers

2k views

### Rolle's theorem in n dimensions

This looks like a statement from a calculus textbook, which perhaps it should be.
"Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that F(a)=F(b) and F'(t) exists ...

**22**

votes

**2**answers

1k views

### Analogues of Luzin's theorem

If $X$ is a compact metric space and $\mu$ is a Borel probability measure on $X$, then the space $C(X)$ of continuous real-valued functions on $X$ is a closed nowhere dense subset of ...

**22**

votes

**1**answer

417 views

### Which ordered fields are homeomorphic to their power?

It is well known that $\mathbb{R}^2\ncong \mathbb{R}$. It is also known that $\mathbb{Q}^2\cong \mathbb{Q}$. It is a corollary to Sierpiński's theorem which states that every countable metric space ...

**21**

votes

**0**answers

520 views

### Relative null-ness

Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer ...

**20**

votes

**11**answers

5k views

### The role of the mean value theorem (MVT) in first-year calculus.

Should the mean value theorem be taught in first-year calculus?
Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that ...

**20**

votes

**4**answers

1k views

### Which is the correct ring of functions for a topological space?

There is a fact that I should have learned a long time ago, but never did; I was reminded that I did not know the answer by Qiaochu's excellent series of posts, the most recent of which is this one.
...

**20**

votes

**3**answers

3k views

### Dual of bounded uniformly continuous functions

Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded uniformly continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$?
I ...

**20**

votes

**2**answers

623 views

### Which smooth compactly supported functions are convolutions?

If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given ...

**19**

votes

**9**answers

3k views

### Function with range equal to whole reals on every open set

There is an example of a function that is unbounded on every open set. Just take $f(n/m) = m$ for coprime $n$ and $m$ and $f(irrational) = 0$.
I want to generalize this in a way to get a function ...

**19**

votes

**2**answers

2k views

### Does Arzelà-Ascoli require choice?

Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version:
...

**19**

votes

**3**answers

952 views

### Is there a function defined on real numbers which is continuous from the left, but not from the right, everywhere

I am teaching Mathematical analysis. A student asked this question. I think this is a good question, but don't know the answer.

**19**

votes

**3**answers

378 views

### Which partitions of $[0,1]$ are collection of level sets of a real continuous function?

Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...