# 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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### 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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### 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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### 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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### $L^1$ 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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### 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.
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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+...
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### 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 ...
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### Hausdorff dimension of graphs .

Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?
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### 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 $F$ ...
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### 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 $k\in\mathbb N-\{0\}$ ...
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### 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 ...
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### 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 ...