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Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

5
votes
1answer
106 views
+50

Sobolev embedding for the $L^q$ norm.

Suppose $f \in H^1(\mathbb R^2)$, where $H^1$ is the Sobolev space, then how to use this information to bound $\Vert f \Vert_{L^q}$, where $q>2$? It seems like Sobolev embedding, but it's not.
5
votes
2answers
184 views
+50

$f(x)/x \to l$ and $f''(x) = O(1/x)$

$f \in C^2(\mathbb{R}, \mathbb{R})$ such that $f(x)/x \to l \in \mathbb{R}$ as $x \to + \infty$, and such that $f''(x) = O(1/x)$ at $+\infty$. Find : $$\lim_{x \to +\infty} f'(x)$$ Some thoughts : ...
3
votes
2answers
86 views
+150

Questions on the real bump function and conjuction of smooth functions

Before I ask you my question which I will mark in bold I will tell you what I already gathered so far. In a previous result I have showed that the bumpfunction is smooth. The bumpfunction is defined ...
12
votes
2answers
328 views
+50

Proof (without use of differential calculus) that $e^{\sqrt{x}}$ is convex on $[1,+\infty)$.

Prove (without use of differentiation) that $f(x)=e^{\sqrt{x}}$ is convex on $[1,+\infty)$. Attempt. Function $x\mapsto e^x$ is convex and increasing, but $x\mapsto \sqrt{x}$ is concave, so we ...
2
votes
1answer
82 views
+100

Shattering with sinusoids

Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$, does there exist an ...
2
votes
5answers
143 views
+50

Evaluating $\lim_{n \to \infty} n\left(1-\frac{1}{e}\left(1+\frac{1}{n}\right)^{n} \right)$

$$\lim_{n \to \infty} n\bigg(1-\dfrac{1}{e}\bigg(1+\dfrac{1}{n}\bigg)^{n} \bigg)$$ If I write expansion of $\bigg(1+\dfrac{1}{n}\bigg)^{n}$ it was equal to expansion of $e$ so $n-n=0$. Is limit is ...
3
votes
0answers
77 views
+50

Surjectivity of real continuous expansive-type functions

Let $f:\mathbb{R}^n \to \mathbb{R}^n$ be continuous and let there exist $\alpha > 0$ such that $||f(\mathbf{x}) - f(\mathbf{y})|| \geq \alpha || \mathbf{x} - \mathbf{y}||$ for all $\mathbf{x}, \...
8
votes
0answers
113 views
+100

Corollary of the Malgrange Preparation Theorem

Let $f:\mathbb{R}\times \mathbb{R}^n \to \mathbb{R}$ be a smooth function, such that $$f(0,0)=0,\ \frac{\partial f}{\partial t} (0,0) = 0,\ldots, \frac{\partial^{k-1} f}{\partial t^{k-1}} (0,0) = 0,\ \...