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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.

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1answer
8 views

For a triangular array, if $\max_{1\le k\le n}x_{n,k}\to 0$ and $\sum _{k=1} ^nx_{n,k}\to \lambda$ does $\prod_{k=1}^n(1+x_{n,k })\to e^{\lambda}$?

Consider the following: we have a triangular array of nonnegative numbers $$x_{1,1 } \\x_{2,1 } \ x_{2,2 } \\ x_{3,1 } \ x_{3,1 } \ x_{3,3 } \\... $$ The maximum on each row converges to zero: $\...
0
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1answer
44 views

Show that if function satisfies $f(kx)=kf(x)$, then $f(x)=\frac{df}{dx}(0)x$

Show that if a differentiable function $f: \mathbb{R}^n\to\mathbb{R}^n$ satisfies $f(kx)=kf(x)$ for all $k$, then $f(x)=\frac{df}{dx}(0)x$. I tried to prove from definition: There exists a linear ...
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0answers
12 views

Examples of increasing continuous function from $\mathbb{R}_+$ onto $\mathbb{R}_+$ satisfying some inequalities

Let $\varphi : \mathbb{R}_+ \to \mathbb{R}_+$ be an increasing homeomorphism satisfying $\varphi(0)=0,$ where $ \mathbb{R}_+:=[0,\infty).$ For example, $\varphi(s)=\frac{s^3}{1+s^2}$ for $s \in \...
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0answers
13 views

prove $ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$

$$ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$$ for all $u \in H^1(\Omega)$ for the open unit circle $\Omega$ in $\mathbb{R^2}$. ...
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0answers
17 views

Best bound for the remainder of two variables Taylor's theorem

Let $f:\mathbb{R}^2 \to\mathbb{R}$ be a two times differentiable function. Then Taylor theorem says that there exists $t\in [0,1]$ such that \begin{align*} f({\bf x})=f({\bf a})+\sum_{i=1}^{2}\frac{\...
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0answers
34 views

Thinking process behind proving $\exists$ solution to $f(x) = x$ when f is bounded

Real Analysis; prove $f(x)=x$ has at least one solution. Here is a solution. I proved initially by doing contradiction, assuming $g(x) = x$ , $\forall x\in$ domain of f, either $f(x) > g(x)$ or $f(...
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0answers
27 views

‎Let ‎$‎c‎>0‎$ ‎and ‎$‎\lim‎ f(x) = L‎$ as ‎$‎x‎\rightarrow ‎+‎\infty‎$‎. Then $‎\lim‎ f(c x) = L‎$ as $‎x‎\rightarrow ‎+‎\infty‎$. [on hold]

‎Let ‎$‎c‎>0‎$ ‎and ‎‎$‎‎‎‎‎\lim‎‎ f(x) = L‎$ as ‎$‎x‎‎\rightarrow ‎+‎\infty‎‎$‎‎‎‎. Then $‎‎‎‎\lim‎ f(c x) = L‎$ as $‎x‎‎\rightarrow ‎+‎\infty‎‎$. Thanks.
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1answer
24 views

About Baire's theorem

Exercise $3.22$ of baby Rudin is to prove the Baire's theorem: Suppose $X$ is a nonempty complete metric space, and $\{G_n\}$ is a sequence of dense open subsets of $X$. Prove Baire's theorem, ...
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2answers
34 views

Prove that $f: X \rightarrow R$ is continuous with respect to the metric $d_1$ on $X$ iff it is continuous with respect to the metric $d_2$ on $X$.

Let $X$ be a set and $d_1, d_2$ be metrics on $X$ so that for constants $m,M > 0$ and any $x,y \in X$ we have $md_1(x,y) \leq d_2(x,y) \leq Md_1(x,y)$ Prove that $f: X \rightarrow \mathbb{R}$ is ...
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2answers
38 views

A problem on number of solutions of a functional equations

Find all functions $f:R \rightarrow R$ such that $f(0)=1$ and for all $x\neq -1$ : $f(x)=8f(2x+1)$ (I have found only one solution: $1/(x+1)^3$. Method was by iterated substitution of $2x+1$ ...
2
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1answer
30 views

Compactness when mapping into a higher $L^p$ space and then back

Question: Let $q>p \ge 1$ and let $T:L^p[0,1] \to L^q[0,1]$ be a bounded linear operator. Let $i: L^q[0,1] \to L^p[0,1]$ be the inclusion map (which is bounded). Is the composition $i\circ T$ ...
0
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0answers
29 views

Uniformly Bounded and Bounded Variation

Studying functions of bounded variation, the following exercise came up: Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of functions with $f_n:I \to \mathbb{R}$. Show that: If $(f_n)_{n\in \mathbb{N}}$...
6
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1answer
245 views

Proof verification: At most countably many local maxima

I'd appreciate a second pair of eyes on a proof. I want to prove that a function $f:\mathbb{R}\to\mathbb{R}$ can have at most countably many strict local maxima. The question has been asked elsewhere ...
0
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1answer
47 views

Differentiability of $|x|^p$?

Let $p > 0$, and let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined piecewise by $f(x)= |x|^p$ if $x \in \mathbb{Q}$ and $f(x)=0$ if $x \in \mathbb{R} \setminus \mathbb{Q}$. For what values of $p$...
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0answers
32 views

Let $f:[a,b] \rightarrow \mathbb{R}$ is cts with $f(a), f(b) < 0, \int_a^b f(x)dx \ge 0$. Show $\exists c \in (a,b) s.t. \int_c^b f(t) dt \ge 0$.

My solution to this problem is the following: Solution: $[a,b]$ is closed interval, so $f$ is uniformly continuous on the interval $[a,b]$. So, we know that there exists $\delta >0$ such that ...
1
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1answer
14 views

Set construction confusion

Let S be an infinite bounded subset of R. Now let's construct a set T where, T={x:x exceeds atmost a finite number of elements of S}. Is there any element of T which is less than the infimum of S? If ...
2
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2answers
39 views

Let $(X,d)$ be a metric space and $ A\subseteq X$ be compact. Prove that for any $y \in X$ there exists $x \in A$ so that $d(y,x) = d(y,A)$

Let $ (X,d) $ be a metric space and $ A \subseteq X $ be compact. Prove that for any $ y \in X $ there exists $ x \in A $ so that $d(y,x) = d(y,A)$ where $ d(x,y)=|x-y| $. Since A is compact it is ...
0
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2answers
34 views

Find the closure and the interior of A

I have to find the closure and the interior of the set A defined by $A =${$(x,\sin(x^{-1})) : x \in R-\{0\}$} $\subset$ $R^2$ I don't know how to start. I know that $\sin(x^{-1})$ has it's maximum ...
0
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0answers
7 views

Uniform convergence and integrals when domain is not a compact set.

Suppose sequence of continuous functions ${f_n}$ that converges uniformly to a continuous function $f$ on a closed interval $[a,b]$, then we have $$\lim_{n\to\infty}\int_a^b f_n(x) dx = \int_a^bf(x) ...
0
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0answers
29 views

If $\sum a_n$ is convergent but not absolutely, then $\sum a_n^+$ diverges

Let $a_n \in \mathbb{R}$, such that $\sum_{n=1}^\infty|a_n|= \infty$ and $\sum_{n=1}^m a_n \to a$, as $m \to \infty$. Let $a_n^+=\max\{a_n,0\}.$ Show that $\sum_{n=1}^\infty a_n^+= \infty$. Approach: ...
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3answers
103 views

If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0.$

Let $g(x)\ge0$. If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0,$ where $f$ is any integrable function. If simeone is allowed to use the Mean Value thorem for integrals, the proof is at hand. ...
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1answer
28 views

Proving Borel measurability of a function.

Suppose $f: \mathbb{R}^2 \to \mathbb{R}$ is a function such that $f(x, \cdot)$ is Borel-measurable for each $x$, and $f(\cdot, y)$ is continuous for every $y$. Define $f_n: \mathbb{R}^2 \to \mathbb{...
1
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1answer
41 views

Where did this definition come from?

Let $C,D > 0$. We call a function $f : \Bbb R \to \Bbb R$ pretty if $f$ is a $\Bbb C^2$-class, $|x^3 f(x)| \leq C$ and $|xf''(x)| \leq D$. (i). Show that if $f$ is pretty, then, given $\...
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1answer
47 views

Why is $\lim_{t\to\infty}\frac{(f(t))^{1+\varepsilon}}{tf'(t)}=\infty$ for all $f:[0,\infty)\to[0,\infty)$ such that $f'>0$? [on hold]

Let $f:[0,\infty)\to[0,\infty)$ be a differentiable function such that $f'(t)>0$ for all $t$ and let $\varepsilon>0$. Why is $$\lim_{t\to\infty}\frac{(f(t))^{1+\varepsilon}}{tf'(t)}=\infty\quad?...
0
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1answer
40 views

Is this product strictly positive?

Let $p\in (0,1)$ and $\varepsilon\in (0, 1)$ be fixed. For all $i\in \mathbb N$ we define $p_i=1+(1-p)(1-\varepsilon)^i$. Is it possible to prove that $$\prod_{i=1}^{+\infty}\frac{p_i}{2-p_i}>0$$? ...
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3answers
55 views

I am stumped on this problem

It is from Intro to Analysis by Bartle 3rd ed Let $I:=[a,b]$ let $f:I\rightarrow\mathbb{R}$ be a continuous function with $f(x)>0$ Prove that there exists a number $c>0$ such that $f(x)\geq c ...
1
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2answers
20 views

Linearity property of integral involving a bounded linear operator

Suppose $X$ is a Hilbert space and $T\in\mathcal{B}(X)$. For $f\in\mathcal{C}([a,b],X)$, where $a\leq b$, we have \begin{equation} Tf\in\mathcal{C}([a,b],X)\quad\text{and}\quad T\int_{a}^{b}f(x)\,\...
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0answers
8 views

Set of roots Hermite polynomials(probabilistic type)

What is the nature of the set of all roots of the Hermite polynomials? They’re known to be all real. Is it a dense set? If not, what are the limit points?Are the limit points also roots? Are the limit ...
1
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1answer
40 views

What is $\int_0 ^1 \int_x^1 \frac{f(t)}{t} dt dx $ if $\int f =1$?

Let $f$ be a Lebesgue integrable function on $[0,1]$ and $\int f = 1,$ and let $$g(x) = \int_x ^1 \frac{f(t)}{t} dt \quad x \in [0,1].$$ Calculate the integral of $g$. I feel like I'm supposed to ...
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0answers
12 views

Sequence of bounded variation functions that converge uniformly to a function with unbounded variation

I need help with this exercise: Prove that $\exists (f_{n})_{n=1}^{\infty}\subset BV([0,1])$ such that $f_{n}\to f$ uniformly, but $\|f_{n}-f\|_{BV}\not\to 0$. So I proposed the sequence $(f_{n})_{n=...
5
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1answer
306 views

What does it mean for something to be strictly less than $\epsilon$ for an arbitrary $\epsilon$?

Perhaps a trivial question, but something I never completely understood. If we have shown that $a-b < \epsilon$ for all $\epsilon > 0$, then does that imply that $a-b \le 0$? I"m interested in ...
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1answer
13 views

Radius of convergence comparison for power series

Given two power series $g (x)=\sum a_n x^n $ and $h (x)=\sum b_n x^n$ with radius of convergence $R_1$ and $R_2$ such that $g(x) \leq h (x)$ for all $x \in \{|x| \leq min \{R_1,R_2\}\}$, Does this ...
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0answers
14 views

The behaviour of functions nested under themselves outwith their domains

This question follows from some interesting observations on a sum of reciprocals. Instead of summing them however, we will place each fraction to make a continued fraction. Some visualisations on ...
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4answers
41 views

How to calculate definite integral of absolute value? [on hold]

How to calculate definite integral of absolute value? How to calculate $\int_0^2|x-1|dx$ for example?
0
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1answer
25 views

Proving sequence converges to a Riemann-Stieltjes integral

Consider the following question: Let $f \in \operatorname{RS}_a^b (\alpha)$, that is, the function $f$ is Riemann-Stieltjes integrable over the interval $[a,b]$ with respect to $\alpha = \alpha(...
5
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1answer
53 views

Convergence of harmonic functions in $L^1$ implies uniform convergence on compact sets

Resorting to an analog of what's done here, I'm trying to prove the following statement: Let $u_m: \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions and suppose there exists a ...
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0answers
20 views

Test if a function is continuous or has at least one discontinuous vertical asymptote between an interval

Imagine evaluating a function with little intervals incrementally across a graph and testing by using the end points of the each interval (and maybe a midpoint), whether the function is continuous for ...
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2answers
35 views

Let $z_1,z_2,z_3 \in \mathbb{C}$ with $z_i=1$ for $i=1,2,3$ and $z_1+z_2+z_3$. Show that $z_i$ are vertices for a equilateral triangle. [duplicate]

Let $z_1,z_2,z_3 \in \mathbb{C}$ with $z_i=1$ for $i=1,2,3$ and $z_{1}+z_{2}+z_3=0$. Show that $z_i$ are vertices for a equilateral triangle. Tip: Think about the case $z_3=1$. What then follows the ...
1
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1answer
25 views

Is this generalization of Minkowski's inequality for sums right?

Could we write $$f^{-1}\left(\sum_{i=1}^nf(|a_i+b_i|)\right)\leq f^{-1}\left(\sum_{i=1}^nf(|a_i|)\right) +f^{-1}\left(\sum_{i=1}^nf(|b_i|)\right)$$ instead of Minkowski's inequality $$\left(\sum_{i=1}...
2
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1answer
55 views

Why can we consider this subsequence?

Why can the sequence be considered as it is? $\{f_n\}$ is Cauchy and $\{f_{n_k}\}$ is a subsequence with the property $\Vert f_{n_k}-f_{n_{k+1}}\Vert_p\le2^{-k}$ On the youtube video https://www....
0
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0answers
27 views

Determine the number of local maxima und minima of this function.

$f:\Bbb R^2 \to \Bbb R:x \to exp(x^2_1+x_2^2)-8x_1^2-4x_2^4 $ Is there any smart way to determine the number of local maxima/minima of this function? We don't neet to find the exact points.
0
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3answers
56 views

Let $f$ be differentiable and $f'>0$ everywhere - Why should $\lim_{x→ ∞} f(x)/f'(x) ≠ 0$?

Let $f$ be a differentiable function $[0, +∞) → [0, +∞)$ and $f'>0$ everywhere. Why should $\lim_{x→ ∞} f(x)/f'(x) ≠ 0$? And I do not know whether this is true. It is my hypothesis.
1
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0answers
23 views

Determining the Values of $\alpha$ for Which the Series is Conditionally and Absolutely Convergent

The task is to determine for which values of $\alpha$ is the following series is conditionally convergent and absolutely convergent. My attempt is below. $$\sum_{n=1}^{\infty} {n^{-\alpha}\cdot(\ln{n}...
5
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0answers
29 views

Does there exist a function $f_{\Box,\Box}(\Box)$ making the formula $a + (b \oplus c) = (f_{b,c}(a)+b) \oplus (f_{c,b}(a)+c)$ true?

Let $a$ and $b$ denote the resistances of two resistors. If they're put in series, the total resistance is $a+b$. If they're put in parallel, the total resistance is $$a \oplus b := \frac{1}{\frac{1}{...
0
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0answers
18 views

Verifying that $x^i e^{-a|x|^2}$ belongs to the Schwartz space

According to wikipedia, $x^i e^{-a|x|^2}$ where $i$ is a multi-index, belongs to the Schwartz space. This means that $$\sup_{x\in\mathbb{R}^n} x^{\alpha} D^{\beta} f(x) < \infty$$ where $f(x) = ...
2
votes
1answer
28 views

Understanding the relation between the dominated convergence theorem and uniform convergence

Problem: Let $f \in L ^ { 1 } , | \widehat { f } | \in L ^ { 1 }$,$$ u ( x , t ) = \int _ { - \infty } ^ { + \infty } \widehat { f } ( \xi ) e ^ { 2 \pi i \xi x - 4 \pi ^ { 2 } a ^ { 2 } \xi ^ { 2 } t ...
1
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2answers
31 views

Proof that $\bigcap_{n\in\mathbb{N}}[a_n,b_n]$ is a non-emtpy set

Let $a_n,b_n \in \mathbb{R}$, for $n\in \mathbb N$ with $a_n \leq a_{n+1} \leq b_{n+1} \leq b_n$. Proof that $\bigcap_{n\in\mathbb{N}}[a_n,b_n]$ is a non-emtpy set. My attempt: Observe $A:=\{a_n : ...
-4
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0answers
37 views

New inequality $(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\sqrt{\frac{\prod_{cyc}(49x-7y+z)}{129(x+y+z)}}\leq \frac{344}{8}$ [on hold]

I'm interested by this problem : Let $x,y,z>0$ then we have : $$(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\sqrt{\frac{\prod_{cyc}(49x-7y+z)}{129(x+y+z)}}\leq \frac{344}{8}$$ I think this problem ...
3
votes
2answers
418 views

Real Analysis: Proof of the equivalent definitions of the derivative.

I am trying to prove to myself that, starting with the definition of the derivative $$f'(x)=\lim_{x\rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}$$ [Note: I wrote the above mistakenly, as pointed out by ...
1
vote
1answer
32 views

Fubini's Theorem - Corollary: product of integrals is the integral of the product.

Let $\phi:[a,b]\rightarrow \mathbb{R}$ and $\psi:[a,b]\rightarrow \mathbb{R}$ integrable functions, then $f:A\rightarrow \mathbb{R}$ defined on $A=[a,b]\times [c,d]\subset \mathbb{R}^{2}$ for $f(x,y)=\...