Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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How to determine whether a multivariate function is bounded or not

Suppose there is a function $f:\mathbb{R}_+^n\mapsto \mathbb{R}$. Are there any systematic ways to determine whether the range of $f$ is bounded or not? For example, there is a function $f(x,y)=-x^2+...
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2answers
653 views

Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ . I recently discovered this result. I am sure it is known, but it is new to me. It is ...
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1answer
2k views

Modified Lebesgue differentiation theorem

Let $\Omega\subset \mathbb{R}^n$ an open set and $u:\Omega\to \mathbb{R}$ be a (locally) $L^1$-function. Then it is well known that the Lebesgue differentiation theorem holds: For almost every $x\in \...
4
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1answer
6k views

Can the supremum of continuous functions be discontinuous on a set of positive measure? [closed]

Given a sequence of continuous functions $f_n(x)$, all defined on a compact set $D$ and assuming $f_n(x)$ is uniformly bounded. Let $f(x) = sup_n f_n(x)$. It is clear that $f(x)$ is not necessarily ...
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0answers
206 views

Elementary analysis: reference request

Given the continuous maps $[0,\infty) \to \mathbb R$ define the following "truncation at level $K$ operator", $T$: $T(f)(t) = f(\min(t, S_f))$, where $S_f = \inf \{ s : f(s) \ge K \}$ So essentially ...
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0answers
215 views

Idea behind choosing $\small f(x)$ as $c^{s}x^{p-1} \frac{[\theta(x)]^{p}}{(p-1)!}$ in the proof that $\pi$ is transcendental [closed]

I am going through the article at this link, where the author proves that: "$\pi$ is $\text{transcendental}$ over $\mathbb{Q}$". Although, I understand the proof, I have some doubts. At page $6$, the ...
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2answers
783 views

$L^1$ norm of the Fourier transform of a truncated Gaussian

I asked this question on Math StackExchange recently but the only useful comment I got was that this could be a good question for Math Overflow. Here it goes: Consider the Gaussian $G(x):=e^{-x^2}$ ...
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0answers
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Almost everywhere differentiability for a class of functions on $\mathbb{R}^2$

A while ago, I came across the following problem, which I was not able to resolve one way or the other. Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and $g(t,...
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2answers
441 views

implicit function theorem for algebraic sets

We know by the standard Implicit Function Theorem that If $f:\mathbb R^4\rightarrow\mathbb > R^2$ is a polynomial (or in fact any continuously differentiable function), then there is a ...
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1answer
167 views

Can symmetrizing a contraction increase the speed of convergence?

Dear community, I have a problem which is very simple to state but seems to be hard to answer. Statement of the problem Let $f$ and $g$ be two symmetric, real functions in $n$ and $m$ variables, ...
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2answers
824 views

reverse mathematics strength of “Lipschitz functions are somewhere differentiable”

What is the reverse mathematics strength of "For all Lipschitz functions $\; f : \mathbb{R} \to \mathbb{R} \;$, $\;$ there exists a real number $x$ such that $f$ is differentiable at $x$." ? (...
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0answers
158 views

minimizing the integral of a function over square sets.

Hi! I'm interested in some problems, but to be honest i'm not sure of the field they belong to. Let $h(x,y)$ be a bivariate function on $X^2$, where $X$ is some nice topological space (for instance $...
2
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1answer
871 views

A singular value inequality

Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$, $s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the singular values of a $2\times2$ matrix. Is it true that $$\left|s_{1}\...
63
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9answers
5k 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 ...
3
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3answers
503 views

Closure of singular points

Let $f(x,y)$ be a complex degree $d$ polynomial that has this particular form. $$ f = \frac{f_{02}}{2} y^2 + \frac{f_{21}}{2} x^2 y + \frac{f_{12}}{2} x y^2 + \frac{f_{03}}{6} y^3 + \frac{f_{40}}{...
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2answers
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Does the Weierstrass function have a point of increase?

Problem The Weierstrass function $W(x)$ is given by $W(x)=\sum_{n\geq 0} a^n \cos(b^n \pi x)$ where $0< a <1$ and $b$ is an odd integer such that $ab > 1+3\pi/2$. A function $f:\mathbb{R}\...
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1answer
401 views

density of a set

let $S=\{\sin (n)|n \in N\}$. We can prove $S$ is dense in $[-1,1]$. So is the set $\{\sin( n^2)|n \in N\}$; but the set $\{\sin (n^3)| n \in N\}$ is not dense in $[-1,1]$. How to prove this?
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1answer
322 views

Convergence Question [closed]

If $\alpha _{n}\rightarrow \alpha$, then how does one show that for any j=1,2,... and $\epsilon> 0$, if $sup\int \left | x \right |^{j+\epsilon }d\alpha _{n}<\infty$, then $\int x^{j}d\alpha _{n}...
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2answers
1k views

How to prove the Hahn-Banach constructively

I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space. Thanks in advance for any helpful answers.
4
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2answers
308 views

Is there a sufficient criteria to guarantee that $\lim_{n} a_{nn} = \lim_{m} \lim_{n} a_{mn}$ ?

Let $a_{mn}$ be a sequence in some $\mathbb{R}^k$. We know in advance that $$\lim_{n} ~a_{nn} = L_1, \qquad \lim_{m}~ \lim_{n} ~a_{mn} = L_2 $$ exist. Is there a sufficient criteria to conclude ...
9
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1answer
948 views

Real analytic function, injective, non surjective and preserving the rationals ?

I'd like to prove the non-existence of a real analytic function, injective, non-surjective that sends rationals to rationals. Is it a classical result ? If not, any hints on how to prove it ? Thanks ...
2
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3answers
841 views

A definite integral

Hello, I am trying to find an explicit form of the following definite integral. I have tried Mathematica and it failed to give an answer. I am wondering whether anyone knows this integral. It might ...
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2answers
1k views

Continuity of a convolution (Version 2)

Hello, This problem bothers me for some time. Suppose that $\mu$ is a non-negative Radon measure (or positive linear functional of the space of continuous functions with compact support); $\psi$ is ...
2
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2answers
2k views

Does a bounded real function have an analytic continuation [closed]

Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where $f$ is real-analytic on the open interval $(0,1)$ $f$ is bounded on the closed interval $[0,1]$ (ie. there is some constant $C$ such that $-...
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1answer
240 views

Conformal Extension from a closed set to open

Let $Q = \{(x,y): x,y\geq 0\} $ be the 1st quadrant of $\mathbb R^2$, and $f$ is a function defined on it such that all the partial derivative(any order) of $f$ exists and continuous. By Whitney ...
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2answers
1k views

An interesting doubly infinite series

Let $0<\mu<1$ and $\alpha:=1-\mu^2$. Consider the function $$f(x):=x\sum_{k=-\infty}^\infty\mu^{4k}e^{-\alpha\mu^{4k}x}-\frac{1}{x}\sum_{k=-\infty}^\infty\mu^{4k}e^{-\alpha\mu^{4k}/x},$$ ...
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0answers
208 views

Monotonicity of a certain parametric integral

I would like to ask for some help (hints, ideas) in solving the following problem: Given integer $n>0$ and real $\alpha>0,\beta>1$ we want to show, that if we define for any $x\in\mathbb{R}...
0
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0answers
294 views

Function space between uniform continuity and Hölder continuity

Can you give an example of a complete metric vector space of uniformly continuous functions that is strictly contained between the set of uniformly continuous functions on $\mathbb R^d$ and the Hölder ...
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0answers
315 views

Gauge integral of the derivative of a function except on a set of measure 0.

For the entire question, the interval I am integrating over is $[0,1]$. Background: In order to exhibit an isometry from $L^2[0,1]$ into $l^2$, I need to either assume absolute continuity over some ...
5
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1answer
503 views

Acceleration via smoothing

Is the following approach to accelerating the rate of convergence of $(1+1/2+\dots+1/n)- \ln n$ (with $n=1,2,3,\dots$), and other sequences like it, in the literature? Let $f(t)=(\sum_{1 \leq n \leq ...
1
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2answers
367 views

A question about zeros of Tate type integral

Fix a positive integer $n$. Fix a continuous character $\chi$ of $\mathbb{R}^*$ with the form $\chi(x)=sign(x)|x|^t$ for some complex number $t$. If $\phi$ is a Schwartz function on $\mathbb{R}$, let $...
7
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2answers
426 views

Multiplying functions on the unit square as generalized matrices

Consider the $\mathbb{R}$-vector space of sufficiently nice real-valued functions on the unit square $I^2$, where "sufficiently nice" could be taken to mean any one of a number of things - say ...
2
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1answer
392 views

Taylor Series Remainder

Suppose I have a $C^\infty$ smooth function $f$ defined on the reals. I can apply Taylor's formula and get the local expression $$ f(x) = \sum_{i=0}^l\frac{f^{(i)}(0)}{l!}x^i+ f^{(l+1)}(\xi(x))x^{l+...
2
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1answer
3k views

Is point to set distance continuous?

Assume $\mathbf{d}:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}_0^+$ is a metric such that the function $\psi(x)=\mathbf{d}(x,y)$ for any $y\in\mathbb{R}^n$ is continuous in the Euclidean ...
4
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1answer
972 views

Hausdorff dimension of graphs .

Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?
4
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1answer
293 views

ordered fields with the bounded value property, without choice

In his answer to my question ordered fields with the bounded value property, Ali Enayat showed that if one assumes the countable axiom of choice, then there exists a non-Archimedean ordered field $F$ ...
15
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1answer
2k views

Interpolating between piecewise linear functions, with a family of smooth functions

Let $[a,b)\subset\mathbb R$, and $F,G:[a,b)\to\mathbb R$ two decreasing piecewise linear functions so that $F(x)\leq G(x)$ for any $x\in[a,b)$. We assume that: there is a number $k\in\mathbb N-\{0\}$ ...
5
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1answer
572 views

Numerically finding a Mercer expansion for a given covariance kernel

Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise. On ...
56
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1answer
3k 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 ...
4
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2answers
2k views

a different nested intervals theorem

Is there any literature on (and a standard name for) the proposition that for any arbitrary-cardinality collection of closed intervals in the reals that is nested (in the sense that, given any two of ...
4
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1answer
215 views

a closed-form for mean/integral, but weighting positive differences between values and “mean” differently from negative differences?

Given a curve $f(x)$ (for $x \in [0,1]$), and a line $y=a$, let $U$ be the total area below $f$ and above $a$, and let $L$ be the total area above $f$ and below $a$. If $L=U$, this means that $a =\...
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2answers
487 views

Estimating the Hausdorff measure of a subset of the sphere

Let $f: S^{n-1}\to \mathbb{R}$ be a continuous function ($S^{n-1}\subset \mathbb{R}^n$ is the unit sphere), $f(a)>0$ and $f(b)<0$ for certain points $a,b\in S^{n-1}$. By continuity these ...
3
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1answer
835 views

ordered fields with the bounded value property

Say that an ordered field $F$ satisfies the bounded value property if, for all $a < b$ in $F$ and for every continuous function $f$ from $[a,b]_F := ${$x \in F: a \leq x \leq b$} to $F$, there ...
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3answers
683 views

truth vs. provability for ordered fields

In Propositions equivalent to the completeness of the real numbers I started by asking "Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of ...
6
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2answers
738 views

A dual theory to the theory of currents?

The k-currents are defined as dual space to the spaces of all smooth k-forms. (These monsters are used to work with the minimal k-surfaces.) Assume I want to look at the generalized k-forms; they can ...
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2answers
568 views

Reference for proof that $C_b^* = rba$

The following theorem seems to have folk status: The topological dual of the space $C_b(X)$ of bounded continuous functions on a topological space $X$ is isomorphic to the space $rba(X)$ of finite, ...
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1answer
706 views

Some infinite products related to prime numbers.

Let $P$ be the set of all odd prime numbers. I am looking for all $s\in(1,\infty)$ for them $ A=\prod_{p\in P} (1+\frac{1}{(p-1)^s})^{p-1} $ exists (i.e. is finite). I know that it should be ...
3
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1answer
405 views

Vanishing on Bad Sets

Let $f: \Bbb{R}^n \rightarrow \Bbb{R}$ be a non-negative function that vanishes on a set $\Omega$ that is compact and has positive measure. What is the minimial amount of regularity required of $f$ to ...
2
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2answers
832 views

Characterization of Weakly measurable functions

I wonder if we can characterize weak measurability of a function taking values in a Banach space using sequence of step functions (functions that have finite range) just like how we define strong ...
5
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2answers
615 views

Darboux function on $[0,1]$ with interesting property

I have proved a few years ago the following proposition: There exists $f: [0,1] \to [0,1]$ with Darboux property such that there exist $A,B \subset[0,1]$ with $A\cap B=\emptyset,\ A \cup B=[0,1]$ ...