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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.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

145439 questions
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Archimedean Property

Show that for every real number $y>0$, $$\bigcap_{n=1}^{\infty} (0, y/n] = \emptyset$$ So this would mean that $0< x \leq y/n$ for every positive integer $n$ which contradicts the Archimdean property?
Tom K
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$C^{1}$ function such that $f(0) = 0$, $\int_{0}^{1}f'(x)^{2}\, dx \leq 1$ and $\int_{0}^{1}f(x)\, dx = 1$

Is there a real $C^{1}$ function on $[0, 1]$ such that $f(0) = 0$, $\int_{0}^{1}f'(x)^{2}\, dx \leq 1$ and $\int_{0}^{1}f(x)\, dx = 1$? I initially was thinking of something like $\pi\sin(\pi x)/2$ or $ce^{x}$ but those satisfy 2 of the 3…
ADF
  • 1,735
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Abbott Understanding Analysis 2nd ed Exercise 2.7.2

I am trying to decide whether the series converge or diverge and am struggling with two different series. I am able to use the comparison test, the absolute convergence test, the alternating series test, and the ratio test. The first series is: $$1…
CUIgrl01
  • 41
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Understanding of the definition of improper integrals

The question is motivated from the following exercise: Prove that $$ \lim_{m\to \infty}\sum_{n=1}^{\infty}\frac{m}{m^2+n^2}=\int_{0}^\infty\frac{1}{1+x^2} $$ Rewriting the series on left hand side, we…
user9464
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Prove that a function is uniformly continuous - without L'Hôpital's rule

Let $f: [0,∞)→[0,∞)$ be a differentiable function, such that $ \lim\limits_{x\to\infty}\big(f(x) + f'(x)\big) = 5$. Prove that $f$ is uniformly continuous. Now, both my tutor and my friends have proved this with a "trick" - multiplying and…
FNB
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Rudin 4.22 Theorem

Could you help me understand why 1. f(H) = B and why 2. $\bar A$ $\cap$ B is empty and why 3. $\bar G$ $\cap$ H is empty?
user86261
  • 685
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Proof that for any $r^3 > 2$ there exists an $h>0$ such that $(r-h)^3>2$

I am trying to prove that for any $r^3 > 2$ there exists an $h>0$ such that $(r-h)^3>2$. I know this is true and I'm trying to prove it using only high school level algebra. So no intermediate value theorems or continuity of real numbers. So far I…
cach1
  • 55
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Real Analysis question second derivative = 0

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be twice differentiable such that $\frac{\partial ^2f}{\partial x\partial y} = 0$. Prove that exists functions $ϕ: \mathbb{R} \rightarrow \mathbb{R}$ and $ψ: \mathbb{R} \rightarrow \mathbb{R}$, twice…
Vitor Hugo Vieira
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Is $\sum_{(n,k)\in\Bbb{N}\times\Bbb{N}}a_{n,k}=\sum_{n\in\Bbb{N}}\sum_{k\in\Bbb{N}}a_{n,k}$ a definition or a theorem?

Let $\{a_{n,k}\}_{\mathbb{N}\times\mathbb{N}}$ a double sequence of positive real numbers. Is the following equality $$\sum_{(n, k)\in\mathbb{N} \times\mathbb{N}}a_{n,k}=\sum_{n\in\mathbb{N}}\sum_{k \in \mathbb{ N }}a_{n,k}$$ a definition or a…
NatMath
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Is the restriction of a diffeomorphism still a diffeomorphism?

Let $g:U \subset {\mathbb R}^n \rightarrow V \subset {\mathbb R}^n$ a diffeomorphism. Is the restriction of $g$ to any subset of $U$ still a diffeomorphism onto its image? I think that it is still a diffeomorphism, because it should be a…
Johny06
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Why is the $y = f(x)$ predicate in the set-builder notation $\{(x, y) | x, y \in \mathbb{R}, y = f(x), x^2 + y^2 = 1\}$ not ambiguous?

I've just started my Bachelor studies in Math, and on my first week in the first quiz there was a question where my answer did not agree with the author's. After a long conversation with my Teaching Assistant, I do not doubt anymore that my answer…
Daniel Pek
  • 143
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Continuous linear extension

Suppose to have a continuous function $f:[a_1,b_1]\to\mathbb{R}$ and let $c_1b_1$. Suppose that you want to extend $f$ to all $\mathbb{R}$ to a new function $g:\mathbb{R}\to\mathbb{R}$ with the following properties: $g$ has to be…
Mathland
  • 526
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Tight bound on least possible norm of $C^k$ functions that interpolate given points

Let $k\ge 2$, and let $f(x):[0,1]\to [0,1]$ have $k$ continuous derivatives. Define the $C^k$ norm of $f$ as— $$||f||_{C^k} = \max_{0\le i\le k} \max_{0\le x\le 1} |f^{(i)}(x)|.$$ Now, given a sequence of $n$ items $(p_n, v_n)$ and given that…
Peter O.
  • 931
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$\left|\int_0^1 \left(\text{sign $m(t)$}\right) \cdot\,f(t)dt\right|<\infty$ but $\int_0^1f(t)dt=\infty$.

Let $m(t)$, $f(t)$ are functions on $[0;1]$ which are assumed to be measurable, $m(t)\neq0$ almost everywhere and $f(t)\geq0$ for all $t$ belongs to the interval $[0;1]$. Let me recall that $\text{sgn}m(t)=\begin{cases}0 \quad\text{if $m(t)=0$}&\\…
Hai Minh
  • 753
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Simple problem of a differentiable function

Please, can somebody help me with this problem? I tried to use the Mean Value Theorem, but couldn't solve it. Let $g: [a,b]\rightarrow\mathbb{R}$ a differentiable function on $[a,b]$. If $g^{\prime}(\theta)\neq 0$, then given $\alpha\in(0,1)$,…
FASCH
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