20 votes
Accepted

In the rational numbers, is every convergent power series a Taylor series for a rational function?

No. Enumerate the rational numbers $a_1,a_2,\dots$. Then for every sequence $c_1, c_2,\dots$ of rational numbers decreasing rapidly enough, the series $$ \sum_{n=1}^{\infty} c_n x^n \prod_{i=1}^{n-1} ...
  • 119k
13 votes
Accepted

If every point is a Lebesgue point of $f$, is $f$ continuous a.e.?

As a warm-up, let's do an example with one point of discontinuity. Our function $f$ looks like this: Here $f : \mathbb R \to \mathbb R$ is zero, except for a sequence of triangular spikes: height $1$...
12 votes
Accepted

A seemingly trivial property of differentiable functions

A counterexample to your assertion: Let $n=1$ and let $$F(x):=x+4^{-j}$$ if $x\in(2^{-j},2^{1-j}]$ for any integer $j$, with $F(x):=x$ for real $x\le0$. Indeed, then $F(x)=x+O(x^2)$ as $x\to0$, so ...
10 votes
Accepted

If a function $f$ is $(1+\varepsilon)$-times Lebesgue differentiable everywhere, is $f$ a constant function?

I realise I'm bumping into you again and already gave you an answer elsewhere after you posted this, but I thought I'd post my answer here for others to see. The answer is yes, $f$ has to be constant ...
8 votes
Accepted

Forcing the uniqueness of a solution of an ODE

$\newcommand\ep\varepsilon$First, the conditions that $f_n\in\mathcal{C}^1([0,1],\mathbb{R})$ and $f_n(x)\ge\sqrt{x}$ for $x\in[0,1]$ imply $f_n(0)>0$. Since \begin{equation*} \begin{cases} y_n(...
6 votes
Accepted

Does this condition on $f$ imply essential boundedness on compacts?

No, this is not true. Let $f \in L^1(\mathbb R)$ and $g_r(x)=\sum_{n=0}^\infty r^n f(x+q_n) \leq \infty$. Then $\|g_r\|_1=\frac{\|f\|_1}{1-r}$ and hence $g_r(x)<\infty$ a.e. If $E_r$ is the null ...
6 votes

Forcing the uniqueness of a solution of an ODE

This question would be possibly at a better place on MathStack Exchange. Yet, once the statement of the question is corrected (the functions $y_n$ need to be defined on $\mathbb{R_+}$ and not only on $...
6 votes
Accepted

Regularity of lipschitz and derivable function

I claim that a function with these properties need not be $C^1$. We start with the function $f: t \in (-1,1)\setminus \{ 0 \} \mapsto \operatorname{sin}(1/t)$, and we also set $f(0) = 0$. The ...
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5 votes
Accepted

Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Yes, there exists such a function: Consider the real line as a linear space over the field $\mathbb Q$ and find a linearly independent Cantor set $C\subseteq \mathbb R$ (using the Kuratowski-Mycielski ...
  • 34.8k
5 votes
Accepted

Does locally nilpotent imply nilpotent for continuous self-maps of intervals?

Yes, this implies that $f$ is nilpotent. As explained in my comment, $f(0)=0$ because otherwise Sarkovski's theorem would give us other periodic orbits which, of course, won't visit $x=0$. We also ...
4 votes

The decay of Fourier coefficients and the continuity of functions

The (distributional) derivative $\mu=f'$ is a measure, and by assumption $\widehat{\mu_n}=o(1)$. By Wiener's theorem, this implies that $\mu$ does not have a point part, so $\mu$ is a continuous ...
4 votes

Regularity of lipschitz and derivable function

Unless I'm overlooking something, you're simply asking whether a function $f:[0,1] \rightarrow \mathbb R$ with a bounded derivative must be continuously differentiable. This can fail quite ...
4 votes
Accepted

Estimate an improper integral

$\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\De}{\Delta}$This inequality does not hold in general, even if the function $f$ is nondecreasing, which will be assumed henceforth. Indeed,...
3 votes

Can we simplify this expression?

Assuming you mean $\epsilon$ is a real positive number and $p$ is a real-valued function in $2+u$ variables, you want a limit of a sum of non-negative numbers, $$\lim_{\epsilon\to0}\sum_{x\in\mathbb ...
  • 51.9k
3 votes
Accepted

Transforming two smooth densities to the same density

This is impossible if $f$ is injective, without further assumptions such as bijective, differentiable, etc. Let $Q_1,Q_2$ be probability measures on a measurable space $(\Omega, \mathcal{F})$, and ...
  • 502
3 votes
Accepted

Does this "local time" type limit exist a.e. for $C^2$ functions?

Even $C^\infty$ isn't enough and even on $\mathbb R$. Take any nowhere dense compact set $A\subset\mathbb R$ of positive measure and set $f=0$ on $A$. Now let $I_k$ be the complementary intervals to $...
  • 53.5k
3 votes

If a function $f$ is $(1+\varepsilon)$-times Lebesgue differentiable everywhere, is $f$ a constant function?

This is only a partial answer: I will give an argument for fixed $\epsilon>0$, assuming that the convergence in the hypothesis is uniform in $x$. Given points $x_0,y$ in $\mathbb R^n$, write $d=|y-...
  • 12.9k
2 votes
Accepted

Boundedness of an extension operator

Let me give an estimate for the first derivatives of $v:=P_1 h$ in terms of the sup-norm of $h$. Setting $z=x'+x_d \xi$ we get $$v(x)=\int_{\mathbb {R}^{d-1}}\phi \left (\frac{z-x'}{x_d}\right )h(z) ...
2 votes

Show that $\frac{1}{n} \sum_{i=1}^n a_i {\rm erf} \left( \frac{b_i-x}{\sqrt{2}} \right) \to x$ for some sequence $\{a_n\}$ and $\{b_n\}$

Let $\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ be the derivative of your $\mathrm{erf}(x/\sqrt{2})$. Then we have $$\sqrt{2\pi}=\int_{-\infty}^\infty e^{-(x-t)^2/2}dt= \frac{1}{n}\sum_{k=-\infty}^\...
2 votes

If every point is a Lebesgue point of $f$, is $f$ continuous a.e.?

Here are some details on Sam Forster’s construction . To make the computation simpler I’d take powers of $4$ instead, i.e. define $g:= \sum_{k=1}^\infty \frac{f_k}{4^k},$ where $f_n$ and $C_n$ have ...
  • 51.9k
1 vote
Accepted

Approximation of Incomplete elliptic integral of first kind

The series expansion in powers of $k$ of the incomplete elliptic integral of the first kind $$F(\varphi,k)= \int_0^\varphi \frac {d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}$$ can be simply obtained by ...
1 vote
Accepted

An inequality about the second-order difference

$\newcommand\de\delta\newcommand\lhs{\text{lhs}}\newcommand\rhs{\text{rhs}}$No. E.g., let $$f(x):=x^3\sin\frac1x$$ for $x\in(0,1/2]$, with $f(0):=0$. Then $f$ can be obviously extended to a ...
1 vote

Axiomatic construction of trigonometric functions

This answer to Geometrically showing $\frac{\alpha}{\beta} > \frac{\sin\alpha}{\sin\beta}$, for $0 < \beta < \alpha < 90^\circ$ shows using only elementary geometry that $\sin \alpha / \...
1 vote

Ekeland's standardness-property inheritable?

The principle for getting examples of nonapplicability of Ekeland's theorem is described by the following simple Example. With $\mathbb I=[0,1]$ consider the Frechet space $E=F=C^\infty(\mathbb I)$ ...
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