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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.

1
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0answers
10 views

Banach Fixed Point Theorem, System Has Solution

Using Banach's Fixed Point Theorem, show that the following system has at least one solution: $$ x = 0.000001x^2 + 10\sin y + 1 $$ $$y = 0.000001y^3 - 0.01\cos x - 1 $$ Here is what I have tried: ...
1
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1answer
19 views

Fourier transform of continuous and integrable function

$f\in L^1(R^n)$,and $f$ is continuous at 0,the Fourier transform of $f$ noted $\widehat{f}$ is non-negative,prove that$ \widehat{f}\in L^1(R^n)$ I have no idea of how to use the condition of ...
4
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1answer
67 views

Decide whether or not $g$ is differentiable at $0$

I need some help with an analysis problem. Define $g:[-1,1]\to \Bbb R$ by $g(x)=(-1)^k/k^2$ for $|x|\in(1/(k+1),1/k], k=1,2,...$ and $g(0)=0$. Decide whether or not $g$ is differentiable at $0$ and ...
1
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0answers
11 views

Show that $\inf \{ \| f-P \|_{\infty}\mid P \in P_n \} \geq \delta_n$ for any decreasing sequence $\delta_n \to 0$

I've been asked to show that given any decreasing sequence $\delta_n \to 0$, we can find a continuous function $f: [-1,1] \to \mathbb{R}$ such that $$\inf\{\|f-P \|_{\infty}\mid P \text{ a polynomial ...
0
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1answer
23 views

What is the final value when moving $x$ by an infinitesimal percentage of $f(x)$ until $100$%?

Let $f : \mathbb{R}\to\mathbb{R}_+$ be a smooth function. Given some $x_0\in\mathbb{R}$ and a direction $s\in\{-1,1\}$, I'm interested in image of $x_0$ under $T_n$ composed with itself $n$ times, i.e....
1
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0answers
21 views

Confusion about Definition of Manifold

I'm reading Munkres' Analysis on Manifolds, and it defines manifold as below: Let $k>0$. A k-manifold in $\mathbb{R^n}$ of class $C^r$ is a subspace $M$ of $\mathbb{R^n}$ having the following ...
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0answers
16 views

Multivariate (Lebesgue) Integration of Composition

$A,B$ are blocks in $ \mathbb{R}^n$, and $f: A \rightarrow B$ a continuous map, such that $ ||f(x)-f(y)|| > c||x-y||$ for all $x,y \in A $ and some $c>0$. If $g: B \rightarrow \mathbb{R}$ is ...
1
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1answer
40 views

Example of function such that integral of $|x(t)|$ is $<+\infty$, and integral of $|x(t)|^2$ is $\infty$

I'm trying to look for an example of a function such that: $$ 1. \displaystyle{\int \limits_{- \infty }^{+ \infty }} \lvert x(t) \rvert \, dt < + \infty $$ $$ 2. \displaystyle{\int \limits_{- \...
3
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1answer
25 views

A Proof about the Oscillation of a Function on Compact Interval

Let $f$ be a bounded function on a compact interval $J$, and let $I(c,r)$ denote the open interval centered at $c$ of radius $r > 0$. Let $\text{osc}(f,c,r) = \sup |f(x)-f(y)|$, where the ...
0
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0answers
9 views

Differentiability of partial function function

Let $E,F$ be Banach spaces. If $f\colon [0,1]\times E\to F$ is continuously differentiable, can we say something about the differentiability of $$g\colon E\to C^1([0,1],F),\quad y\mapsto f(\cdot,y)?$$ ...
1
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0answers
16 views

How to show the following weak convergence using characteristic functions

Suppose $g:R\rightarrow R$ has at least three bounded continuous derivatives and let $X_i$ be $iid$ and in $L^2$. Prove that: $\sqrt{n}[g(\overline{X_n}) - g(\mu)]\xrightarrow{w} N(0,g^{'}(\mu)^{2} ...
4
votes
0answers
34 views

$f$ is Riemann integrable if the set of discontinuities is measure zero.

Let $f$ be a bounded function on a compact interval $J$, and let $I(c,r)$ denote the open interval centered at $c$ of radius $r>0$. Let $osc(f,c,r)=\sup|f(x)-f(y)|$, where the supremum is taken ...
0
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2answers
28 views

Problem with the proof of $g^{-1}(A\cap B)\subseteq g^{-1}(A)\cap g^{-1}(B)$.

This is Abbott's exercise $1.2.7.b$. and it is not a duplicate of the following questions. Proof of $f^{-1}(A) \cap f^{-1}(B)= f^{-1}(A \cap B)$ how to prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap ...
1
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1answer
17 views

Interior solution of a Variational Inequality in Hilbert Space

I'm trying to understand the proof of the following claim: Let $K \subset \mathbb{R}^n$ be compact and convex and let $$F:K \rightarrow (\mathbb{R}^n)'$$ be continuous. Then there exists an $x\in K$...
3
votes
2answers
252 views

Limits of Rolle theorem

I would like to see a function $f:[a,b]\to\mathbb{R}$ that is differentiable in $(a,b)$ but it is not continuous at least at one of the interval boundary points $a$ or $b$. Can you show me one? This ...
1
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1answer
51 views

Solving problems regarding $L(x)=\int_1^x\frac{dt}{t},\quad x>0$ and the inverse function $E(x)$

The problem Without using $L(x)=\ln(x)$ and given $$L(x)=\int_1^x\frac{dt}{t},\quad x>0$$ a) Prove that i) $L(xy)=L(x)+L(y),x,y>0$ ii) $L(1/x)=-L(x)$ b) Prove that $L(2)<1$ c) Prove ...
0
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1answer
21 views

Finding the limit of a sequence of functions formally

Let $$g_n:\mathbb{R}\to\mathbb{R}$$ $$g_n(x)=\begin{cases} 1 & \mbox{for }x\in[-n, n] \\ x+n+1 & \mbox{for }x\in[-n-1, -n] \\ -x+n+1 & \mbox{for }x\in[n, n+1] \\ 0 & \mbox{otherwise} \...
20
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2answers
1k views

mystery regarding power series of $\frac{1}{\sqrt{1+x^{x}}}$

In the course of playing around with $\sum_{n=1}^{\infty} \frac{1}{\sqrt{1+n^{n}}}$, I used w|α to obtain the power series for $f(x)=\frac{1}{\sqrt{1+x^{x}}}$ which is \begin{align*} \frac{1}{\sqrt{...
1
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0answers
55 views

Formal group law and Koenigs function conjecture !?

Let $f(x,y)$ be a symmetric real function and a formal group law $$G(x + y) = f(G(x),G(y)). $$ Consider the equation $$ h(2x) = f(h(x),h(x)) = A(h(x)). $$ This equation has many solutions. ...
2
votes
1answer
32 views

Proof of L'Hospital's Rule on Zorich book: an unclear step.

The proof of L'Hospital's rule on Zorich, Mathematical Analysis I, chapter 5.4, pag. 251, starts with: if $g'(x)\neq 0$ we conclude on the basis of Rolle's theorem that $g(x)$ is strictly monotonic ...
0
votes
3answers
75 views

Is this function bounded on $[a,b]$?

I need to show that $f:[0,1] \rightarrow \mathbb{R}$ is bounded on $[a,b]$ if for all $c \in [a,b]$, $\lim\limits_{x \rightarrow c} f(x)$ exists. I tried using the definition of limit, but I don't ...
0
votes
0answers
12 views

Extensions of Riemann-Stieltjes without continuity problems

Are there any extensions of Riemann-Stieltjes integration that are able to handle the following integral? $\int_0^1 \alpha \space d\alpha$ where $ \alpha(x) = \left\{ \begin{array}{lr} 0 & ...
0
votes
0answers
43 views

continuous but non-differentiable function having countable points of non-differentiablity [on hold]

Consider function $f:\mathbb{R}\to\mathbb{R}$ satisfying $|f(x)-f(y)|\lt 4321|x-y|$ for all real numbers $x$ and $y$. Show that there exists a function $f(x)$ such that $f$ is continuous but is non-...
0
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0answers
10 views

Optimal Control: Proof of Adjoint Lagrange method?

Consider an optimal control problem: Let $x_0\in\mathbb{R}^n$, $f\in C^{0,1}([0,T]\times (\mathbb{R}^n\times \mathbb{R}^m),E)$ be bounded, let the state equation be $$\dot{x}(t) = f(t,x(t),u(t)),\, t\...
1
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0answers
25 views

Finitely Additive Homogeneous Translation Invariant Measure on $\mathcal{P}(\mathbb{R})$

IMPORTANT EDIT: this question has been moved to Mathoverflow It is known that there exists a finitely additive translation invariant measure on $\mathbb{R}$ that extends the Lebesgue measure. I.e. a ...
1
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0answers
27 views

Question on a step of the proof of Theorem 1.25 of Introduction to Fourier Analysis on Euclidean Spaces

Blockquote Theorem 1.25: Suppose $ \phi \in L^1(\mathbb{R^n}) $ and $ \int_{\mathbb{R^n}} \phi =1 $ . Also, let $\phi_{\epsilon}(x)=\frac{\phi(\frac{x}{\epsilon})}{\epsilon^n}$.Moreover , suppose ...
0
votes
1answer
30 views

Differentiability implies continuity proof

Statement: If $f: \mathbb{R}^n\rightarrow \mathbb{R}^m$ is differentiable at $\underline{x}$, it's continuous at $\underline{x}$. My proof: Since $f$ is differentiable at $\underline{x}$, all ...
8
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5answers
137 views

If $‎\lim\limits_{x\to\infty}‎\frac{f(x)}{g(x)} = 1‎$,‎ then $\lim\limits_{x\to\infty}(f(x) - g(x)) = 0‎$.

‎Suppose ‎$‎f‎$ ‎and ‎‎$‎g‎$‎ ‎‎are real functions such that $‎‎‎\displaystyle{\lim_{x\to\infty}}‎\frac{f(x)}{g(x)} = 1‎$‎‎. ‎My ‎question ‎is‎: ‎‎‎ ‎‎What other condition is required ‏‎that $‎‎‎\...
0
votes
2answers
29 views

Proof of a compact set without using Heine-Borel

I'm struggling with a question about compact set, but I can't figure out it on my mind. Is this set compact or not? Prove it without using Heine-Borel Theorem. a)$\mathbb{R}^n$ such that $B$ doesn't ...
2
votes
1answer
43 views

The set at which a function's oscillation is at least $\epsilon$ is closed

Let $f$ be a bounded function on a compact interval $J$, and let $I(c,r)$ denote the open interval centered at $c$ of radius $r>0$. Let $osc(f,c,r)=\sup|f(x)-f(y)|$, where the supremum is taken ...
0
votes
0answers
15 views

Continuous dependence of first Dirichlet Eigen value on the domain.

Let $\Omega$ be a open, bounded domain in $R^n$. Let $Lu=\sum (a_{ij}(x)u_{x_i})_{x_j} +c(x) u$ be the operator where $a_{ij}=a_{ji}$ are in $C^1$ and $c(x)$ is continuous. Let $\Omega_0$ be a bounded ...
1
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0answers
66 views

If $u^3+v+w=x+y^2+z^2,u+v^3+w=x^2+y+z^2,u+v+w^3=x^2+y^2+z,$ prove the following

I am stuck with the following problem that says : If$$\begin{align} u^3+v+w&=x+y^2+z^2,\\u+v^3+w&=x^2+y+z^2,\\u+v+w^3&=x^2+y^2+z, \end{align}$$then prove that $$\frac{\partial(u,v,w)...
0
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0answers
21 views

Kolmogorov's 0-1 law with indicator function.

Let $A_1, A_2, \ldots$ be any independent sequence of events and let $S_x := \{ \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} 1_{A_n} \leq x \}$. Prove that for each $x \in \mathbb{R}$ we have $\...
2
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1answer
89 views

Consider $F(a)=\int_{0}^{+\infty}{\frac{1}{1+x^a(\ln(1+x)^2f(x)}} \, \mathrm{d}x$

Let $f$ be $T$-periodic, continuous, such that $T=1$, $f(0)=0$, $f'(0)>0$, and $f(x)>0$ for $x\in(0,1)$. $$F(a) = \int_{0}^{+\infty} {\frac{1}{1+x^a(\ln(1+x))^2f(x)}} \, \mathrm{d}x$$ Consider ...
2
votes
2answers
88 views

Provide example of two series that both diverge but $\sum\min\{a_n,b_n\}$ converges

I've posted the solution for this problem and I'm trying to understand this. In the end of the solution provided it says to continue this process. So, do we hold $a_n$ to be $\frac{1}{n^2}$ and $b_n$...
1
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0answers
16 views

Statement of Riesz-Thorin theorem

I've seen a number of sources (most I have seen I think) state the Riesz-Thorin theorem in the following way: Let $(\Omega_{1},\Sigma_{1},\mu_{1})$ and $(\Omega_{2},\Sigma_{2},\mu_{2})$ be $\sigma$-...
2
votes
0answers
41 views

Why does evaluation of a two-variable limit fail when using polar coordinates?

The definition of the limit of a two-variable function: $\lim\limits_{(x,y)\to (a,b)}f(x,y)=L\,$ if and only if for all $\epsilon>0$ there exists a $\delta >0$ such that $$0<\sqrt{(x-a)^2+...
0
votes
0answers
62 views

Comparing $\sigma$ algebras with topologies

I am learning measure theory from Papa Rudin. I am just trying to capture the ideas conceptually. First of all, how far is measurable sets from topological sets. I guess this question could be ...
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.
1
vote
0answers
26 views

Homeomorphism from the real numbers to the real numbers with restriction to the Cantor set.

Let $K$ be the Cantor set and $C \subset R$ be a non empty compact set with no isolated points and empty interior. Prove that it exists a homeomorphism $f:R \longrightarrow R$ such that $f(K)= C$.
2
votes
1answer
37 views

Evaluate Infinite Series Summation of 1/(k(k+c)) [duplicate]

I am trying to evaluate the infinite series: $$\sum_{k=1}^{\infty} \frac{1}{k(k+c)} $$ where c is a constant, positive integer. Is my approach correct by using partial fraction decomposition? Then ...
3
votes
2answers
425 views

Product rule for matrix-valued functions and Differentiability of matrix multiplication

Let $F, G: \mathbb{R}^{n \text{ x } n} \to \mathbb{R}^{n \text{ x } n}$ be two differentiable functions. Defining $(F \cdot\ G)(A):=F(A)G(A)$ using the usual matrix multiplication, prove that $F\cdot\ ...
0
votes
1answer
27 views

How to prove the finiteness of lower semi-continuous function?

Let $C$ be a convex set, and let $f$ be a lower semi-continuous convex function. In addition, we have $f$ is bounded above on every bounded subset of $\text{ri}~C$, where $\text{ri}~C$ is the relative ...
0
votes
0answers
25 views

Quick way to compute $\int_{0}^{\infty} f(x) g(x+y)\, dx$ for $y \in [0,\infty)$

When we want to compute: $$ h(y) = \int_0^y f(x) g(y-x)\, dx, $$ for $y \in [0,\infty)$ we can make use of a Fast Fourrier Transform to quickly compute these integrals. I have a similar problem where ...
0
votes
1answer
83 views

Help required to prove the following inequality… [on hold]

Our professor wrote this problem on board and left the room stating, "Try it if you want. You'll succeed only if have properly gone through the concepts that were taught last week." Let $f :[0,1]\...
6
votes
1answer
146 views

Algebraic proof that two statements of the fundamental theorem of algebra are equivalent

Students studying the fundamental theorem of algebra in high school are probably familiar with the statement that goes something like the following. Every non-zero, single-variable, degree $n$ ...
10
votes
1answer
107 views

Let $S=\{p(x) \in \mathbb Z[X] :|p(x)| \leq 2^x, \forall x\in \mathbb N\}$. Prove $|S| < \infty$

Question: Let $S=\{p(x) \in \mathbb Z[X] :|p(x)| \leq 2^x, \forall x\in \mathbb N\}$. Prove $|S| < \infty$. Notice this is not true in $\mathbb R[X]$, as $|x-a|\leq2^x $, $a\in[0,1]$ shows. ...
0
votes
0answers
13 views

Measure in the square with uniform marginals are characterized by their support?

Is it true that two measures on the unit square with uniform marginals and the same support are the same measure? If not, can you show an easy counterexample?
3
votes
1answer
727 views

Unique weak solution of Poisson's equation

Let $\Omega$ be an open set in $\mathbb{R}^n$ and now consider the weak formulation of Poisson's equation $$\int_{\Omega} \langle Du,Dv \rangle = \int_{\Omega}{fv}$$ for $v \in H_0^1$ and $u \in H_0^1$...
0
votes
0answers
15 views

How can we apply the non-stationary phase theorem with dependence on large parameter?

I would like to estimate the following integral for long times $T>0$ independent of $\epsilon$ in the limit that $\epsilon \rightarrow 0$ $$G_\epsilon(t) := \frac{1}{\epsilon}\int_0^t f(s,t) e^{\...