# 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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### Differentiability of the blow-up of a function

Let $u \in C^0([-1,1])$ such that $u(0)=0$. Suppose that $u$ satisfies the following property: For every $\{t_k\}\subset \mathbb{R}$ such that $t_k \to 0$, there exist a real number $\alpha$ (...
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### Pointless characterization relating between a fractal and its code space

Given an hyperbolic IFS $(X,\{f_i:i=1,\ldots,N\})$ and denoting its code space by $\Sigma_N = \{1,\ldots,N\}^{\mathbb{N}}$ and the generated fractal set by $\mathcal{A}$. There is a continuous ...
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### Poincaré lemma for gradient times its transpose

Poincaré lemma states that a vector $v_i(x)$ defined on a ball in $R^n$ is the gradient of a function if and only if \begin{equation} \partial_i v_j = \partial_j v_i \end{equation} or equivalently ...
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### On a case of real-analytic interpolation

Given strictly increasing sequence $x_n$ of rational numbers with $\sup x_n = x$. In which case (sufficient condition on $x_n$) there exists real-analytic function $f:U_\epsilon(0)\to\mathbb{R}$ ...
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### Real-analytic function with given set of values [closed]

We say that a strictly increasing sequence $x_n$ of reals converges fast to $x$, if for each $k\in\mathbb{N}$ the sequence $n^k\cdot(x_n − x)$ is bounded. It is known that there exists a $C^\infty$-...
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### Does the average primeness of natural numbers tend to zero?

This question was posted in MSE. It got many upvotes but no answer hence posting it in MO. A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value ...
170 views

### If the Hausforff dimension of the graph of a function $u$ is $N$ and $\tilde u = u$ a.e. then $\dim_H \mathrm{graph} \, \tilde u = N$ too

Let $\Omega$ be an open (non empty) set and $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a function such that the Hausdorff dimension of its graph is $N$. Let $\tilde u = u$ a.e. Is it true ...
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