# Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property and related analysis topics, such as continuity, differentiation, and integration.

138,677 questions
Filter by
Sorted by
Tagged with
50 views

### Stein's proof of Lusin's theorem: Is the $E_n$ necessary?

I wonder if this is a simpler and correct proof of the Lusin theorem stated below. I was inspired by Stein's proof. However I think the $E_n$ in Stein's proof may be unnecessary. This is the theorem ...
• 409
52 views

• 3,128
24 views

### Topological conjugacy for vector field failing on the level of flows

Given two vector fields $F_1,F_{2}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with their flows $\phi_{1}, \phi_{2}:\mathbb{R} \times \mathbb{R}^{n}\rightarrow \mathbb{R}^n$ We say they are topological ...
• 417
14 views

### Recursive sequence from piecewise-linear interpolation of $x\mapsto x^{p/q}$ for $p/q\in(0,1)$ only seems to converge to it if $p/q=1/2$

So, recently i got a bit (too) curious about approximating $x\mapsto x^{p/q}$ for arbitrary values of $p/q\in{]0,1[}$, starting with the following piecewise linear interpolation for all integer $n\ge1$...
78 views

120 views

### An asymptotic expansion for $u_{n+1} = u_n^2 + u_n$

I'm completely stuck finding an equivalent of the following recurring sequence. $\left\{\begin{matrix} u_0 > 0\\ u_{n+1} = u_n^2 + u_n \end{matrix}\right.$ The problem suggests to use the sequence ...
• 101
44 views

### Convergence of the sequence $\sum \frac{n+5}{\sqrt{n^5 - 10n+10}}$

I am stuck on a question and would really appreciate if anyone can give some help. Determine whether the series: $$\sum_{n=1}^{\infty} \frac{n+5}{\sqrt{n^5 - 10n+10}}$$ converges or not. I am trying ...
1 vote
36 views

### Show that $f'(y) = 0$ when $f$ has a local maximum at $y$ [duplicate]

Suppose $f: [0,1] \to \mathbb{R}$ is a differentiable function. For some given $y \in (0,1)$, $\exists$ $\epsilon > 0$ such that $$f(x) \leq f(y) \ \forall \ x \in (y-\epsilon, y+\epsilon).$$ Show ...
• 147
37 views

• 846
1 vote
32 views

### Brezis's Proposition 4.21: did the author mean $f \star \rho_n$ rather than $\rho_n \star f$?

For $f \in L^1\left(\mathbb{R}^N\right)$ and let $g \in L^p\left(\mathbb{R}^N\right)$ with $1 \leq p \leq \infty$, we define $$(f \star g)(x)=\int_{\mathbb{R}^N} f(x-y) g(y) d y.$$ A sequence of ...
• 13.8k
1 vote
25 views

### The function $\rho$ defined by $\rho (x) := \begin{cases} e^{1/ (|x|^2-1)} &\text{if } |x| <1 \\ 0 &\text{otherwise} \end{cases}$ is smooth

We define $\rho:\mathbb R^n \to \mathbb R$ by $$\rho (x) := \begin{cases} e^{1/ (|x|^2-1)} &\text{if } |x| <1, \\ 0 &\text{otherwise} \end{cases}$$ It's is mentioned at page $108$ of ...
• 13.8k
1 vote