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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2
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0answers
19 views

Does $\int_V f(s)ds=\int_U f(\varphi (t))|\det \varphi '(t))|dt$ hold if $f$ is measurable?

Let $\varphi :U\to V$ a diffeomorphism where $U,V$ are open. If $f:\mathbb R\to \mathbb R$ is a Borel function integrable function, we have that $$\int_V f(s)ds=\int_U f(\varphi (t))|\det \varphi '(t)|...
0
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0answers
4 views

How to determine strip condition of nonlinear PDE?

I started learning PDE on my Own. I was doing one example in the book but I stuck at one step. I do not understand how the Author come at the conclusion about strip condition. I understand everything ...
0
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1answer
18 views

Re: Construction of a non-empty perfect set of real numbers without rationals

We consider the set $[e, \pi]$. Let $\{x_1,x_2,x_3,...\}$ be the enumeration of the rationals in $[e, \pi]$. $r_n= \min\{|x_n-e|,|\pi-x_n|\} $ We can enclose $x_n$ by $I_n=(a_n,b_n)=(x_n-\frac{r_n}{...
1
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0answers
13 views

Saddle point method for a stationary subspace of dimension $> 0$

I recently asked a question on Physics SE regarding the validity of using the saddle point technique when the saddle point is not only degenerate, but forms a continuous subspace of the parameter ...
0
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0answers
22 views

Let $f$ is function from $\mathbb{R}^n$ to $\mathbb{R}$, such that it is convex by every variable (if all another fixed).

Prove that $f$ is continuous. It probably correct, that $f$ has continuous derivatives by every variable. But also, for example, $f_{xy}$ can no exist. Hence, need to take definition of continuous.
-2
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1answer
28 views

Complex plane minus origin is open?

How to show that $\mathbb C \setminus \{0\}$ is an open set? The definition of an open set is: A subset $E$ of $X$ is said to an open set if its all points are interior points.
10
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0answers
44 views

A nice Combinatorial Identity

I am trying to show that $\forall N\in\mathbb{N}$, $$\sum\limits_{n=0}^{N}\sum\limits_{k=0}^{N}\frac{\left(-1\right)^{n+k}}{n+k+1}{N\choose n}{N\choose k}{N+n\choose n}{N+k\choose k}=\frac{1}{2N+1}$$ ...
-6
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0answers
30 views

Can I find a K such that $\int_{0}^{1}f^2(x)dx<K(\int_{0}^{1}f(x)dx)^2$? [on hold]

Can I find a $K$ such that $$ \int_{0}^{1}f^2(x)dx<K(\int_{0}^{1}f(x)dx)^2\:\:? $$ $K$ can be a constant or another function. I dont know how to approach the problem. I want to know if there ...
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0answers
16 views

Bounded sequence of functions issue [on hold]

I was stuck in a problem as follows. Find a bounded sequence $\{f_m\} \subset C(I) = \{f$ is continuous on $I|f: I \rightarrow R^n, I = [a, b] \subset R\}$ such that there is no convergent ...
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0answers
7 views

IVP with locally defined solution in $C^{1}$

Let $J\subset \mathbb{R}$ be an open interval. Lets suppose that the IVP $\dot x = f(t,x)$, $x(t_{0})=x_{0}$ with $f\in C(J \times \mathbb{R})$ and $(t,x)\in J \times \mathbb{R}$ has a solution in $C^{...
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3answers
32 views

Let $g(x)= \sup_{n}f_n(x)$. Prove that $g^{-1}((a,\infty]) = \cup_{n=1}^{\infty} f_n^{-1}((a,\infty])$

I am doubtful of my solution. So, I wanted to make sure. Let $g,f_n: X \rightarrow [-\infty, \infty], \forall n \in \mathbb{N}$ s.t. $g(x)= \sup_{n}f_n(x) \; \forall x \in X$. Prove that $$ g^{-1}((a,...
0
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2answers
22 views

Finding example of a function having the required property.

Does there exist any continuous function $f : \Bbb R \longrightarrow \Bbb R$ which is not differentiable only at the integers and is not uniformly continuous everywhere? The only function which I can ...
0
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2answers
48 views

Prove that $a<b$ iff $a^3<b^3$ (don't use difference of two cubes)

I would like to know a different proof for this problem: Let $a,b\in \Bbb R$. Show that $a<b$ iff $a^3 < b^3$. Given that it is not much difficult to prove by using a difference of cubes and ...
-1
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0answers
17 views

eigenvalue distribution

Assume a set of $n$ eigenvalues of some specific $n \times n$ matrix given by $\lbrace\lambda_{1},...\lambda_{n}\rbrace$. I have an expression for the second smallest eigenvalue $\lambda_{2}$ $$ \...
1
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1answer
20 views

Interpretation of solution of pde of form $f(u,v)=0$

Given PDE $zz_x-zz_y=z^2+(x+y)^2$ I had solved problem and found that solution is $F(x+y,\log(x^2+y^2+z^2+2xy)-2x)=0$ I know how to solve Above type of PDE. But problem is that I do not know ...
1
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1answer
33 views

If $g(f(x))$ is continuous if and only if $g$ is continuous, then $f$ is a homeomorphism

Let $f : M \rightarrow N$ and $g : N \rightarrow \mathbb{R}$, where M and N are metric spaces. Let $f$ be one to one and onto. Suppose $g$ is continuous if and only if $g\circ f$ is continuous. I ...
1
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1answer
20 views

Prove that a mapping from a separable space to the Hilbert cube is a homeomorphism

Suppose $(M, \rho)$ is separable, and that $\rho(x,y)\leq 1$ for all x and y in M. Let $x_n$ be a countable dense set of M. Define the Hilbert cube $H^{\infty}$ as the collection of all real sequences ...
2
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0answers
52 views

Is fractional part of $e^n$ dense in (0,1)?

Can fractional part of $e^n$ where $n\in\mathbb{Z}_+$ dense in $(0,1)$? Generally, for which kind of $\alpha > 0$, $\{\alpha^n\}$ dense in $(0,1)$?
1
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1answer
30 views

Difference between $L^2$ and $L_{\sigma}^{2}$

Defining the set Let $n \geq 2$, and let $\Omega$ be a domain in $\mathbb{R}^{n}$. Define the set $ C^{\infty}_{0,\sigma}(\Omega) := \{ f \in C^{\infty}_{0}(\Omega)^{n} \ | \ \text{div}(f) = 0 \}$. ...
0
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1answer
72 views

Can a function have multiple limits?

Is this proposition true? If $A$ is the limit of function, when $x$ approach to $y$, according to the definition of limit by Cauchy, the number which have the difference with $A$ is smaller than any ...
0
votes
1answer
29 views

Interchange max and limit

Suppose a sequence of continuous functions $f_n(x)$ converges to a function $f(x)$ on $x \in \mathcal{C}$ pointwise, i.e., $$\lim_{n \to \infty} f_{n}(x) = f(x), \ \forall \ x \in \mathcal{C}.$$ Then ...
3
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2answers
46 views

Find an open set which has finite length and is super set of rational numbers

We define the length of the open set as the difference between the ends. $l[(a,b)]=b-a$ and $l[(a,b)\cup(c,d)]\leq l[(a,b)]+l[(c,d)]$ We have to find one open set $U$ such that $l(U)<\infty$ and ...
0
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1answer
32 views

Is the form domain of a self adjoint operator closed?

If $\mu$ is a positive, finite measure on $\Bbb R$, define $Q=\{f\in L^2(\Bbb R,\mu): \int_{\Bbb R} |x| |f|^2\text d\mu<\infty\}$ and $q(f,g)=\int_{\Bbb R} x\bar f g\text d \mu$. Suppose there ...
1
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1answer
16 views

Proving limit of bounded convergent sequence is bounded

Question: Show that if $a \leq x_n \leq b$ for every $n$ and $x_n \rightarrow x$, then $a \leq x \leq b$. Proof: Let $\epsilon>0$. By assumption $a_n \leq x_n \leq b$ for all $n$. By definition ...
0
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1answer
26 views

Continuous function with compact support

Let $1 \leq m \leq n$, $g$ and $h$ continuous function of compact support, $g: \mathbb R^m \to \mathbb R$ and $h:\mathbb R^{n-m} \to \mathbb R$. We define $g \star h: \mathbb R^n \to \mathbb R$ such ...
5
votes
1answer
58 views

$ \sum_{k=1}^{\infty}\frac{\log(k)}{k^2(k+1)}$ explicit value.

I am trying to evaluate the following series explicitly, $$ \sum_{k=1}^{\infty}\frac{\log(k)}{k^2(k+1)}$$ I know this converges by the comparison test. I have tried defining a function, $$f(t)=\sum_{...
0
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1answer
39 views

Solving Basel Problem using euler infinite product and infinite product-sum equality

Trying to prove Basel problem through the equality $sin(x) = \prod\limits_{k=1}^{+\infty}(1-\frac{x^{2}}{\pi^{2}k^{2}})$, I came across the following problem; I was able to prove the following ...
3
votes
1answer
59 views

Well ordered subsets of $\mathbb{R}$

At the entrance exam of a french school, the following problem was given : Characterize well-ordered subsets of $\mathbb{R}$ The only property I found was that such a subset must be at most ...
0
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1answer
38 views

How to tell whether a function has a root in a specific set?

There is a smooth function $$f:\mathbb{R}\rightarrow\mathbb{R}$$ Suppose that we know everything about behaviour of this function $f(x)$ for $x\ge M$ where $M$ is real number. To know everything i ...
1
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1answer
29 views

Study the uniform convergence of the functional sequence $f_n(x)=\sqrt{n+1}\sin^nx\cos x$

Study the uniform convergence of the functional sequence $f_n(x)=\sqrt{n+1}\sin^nx\cos x$ I found the limit $$\lim_{n\to \infty}f_n(x)=0$$ The solution in the book it says that since $$d_n=\sup_{...
1
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1answer
58 views

What is the value of $\frac{1}{2\pi}\int_{-\infty}^{\infty}\prod_{k=1}^\infty\Big(1-p+p\exp \frac{it}{2^k}\Big)dt$?

Here $0<p<1$. If $p=\frac{1}{2}$ the value is $1$. In all cases the value is a positive real number (or $+\infty$). This integral is associated with the inverse Fourier transform of some ...
0
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2answers
18 views

Pointwise Convergence of a Uniformly Bounded Sequence of Functions

I have a sequence of functions $f_n:[0,1] \rightarrow \mathbb{R}^n$ such that $f_n$ is uniformly bounded, i.e. $\|f_n\|\leq M$ with $M$ independent of $n$. Is it true that that the sequence $(f_n)$ ...
2
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4answers
167 views

Circular Reasoning for Epsilon-Delta Proof?

$$\lim_\limits{x \to 4} 2x-5=3$$ In order to prove this limit, the epsilon-delta definition will be used. $$|f(x)-L|<\varepsilon$$ $$|x-a|<\delta$$ In the proof, the above $2$ inequalities ...
0
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1answer
38 views

“every regular functions is differentiable infinitely” [on hold]

Is there a theorem that states something like "every regular functions is differentiable infinitely"? I want to prove that $$\frac{1}{1+x^2}$$ (or similar "simple" functions) has a second derivative ...
1
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1answer
62 views

If $f(x)=0$ for all but countably many points in $[a,b]$

First question in text is 'Let $f$ be real-valued function on $[a,b]\ s.t.$ $f(x)=0$ for all $x\neq c_1, ... , c_n.$ Prove that $f$ is Riemann-integrable with $\int_a^bf =0 $.' I proved this question ...
2
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0answers
20 views

Finite union of intervals as finite union of disjoint intervals

From "Principles of Mathematical Analysis" by Walter Rudin: Let $R^{p}$ denote $p$-dimensional euclidean space. By an interval in $R^{p}$ we mean the set of points $\vec{x} = (x_{1}, \ldots, x_{p}...
0
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0answers
32 views

Statement about the occurrence of a digit in a decimal seems false.

Let $x$ be a decimal expressed in the scale of $r$. Meaning, if $x=\frac{a_1}{r}+\frac{a_2}{r^2}+...$ where $0\leqslant a_i\leqslant r-1$, then the decimal expressed in the scale of $r$ is $0.a_1a_2......
0
votes
1answer
26 views

Can the sum property of integrals be stated for indefinite integrals?

Some time ago I learned about the following property of integrals: If $ f $ and $ g $ are bounded, integrable functions on $\color{red}{[a, b]}$, then so is $f + g$ and $$ \displaystyle \int_\...
1
vote
1answer
15 views

Struggling to understand why two different definitions of Baire Category theorem are same

I have two versions of Baire Category theorem and I am struggling to find why they are equivalent: My professor notes says " Any complete metric space is of second category i.e. we cannot write it as ...
0
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4answers
88 views

Strange! Right solution of equation is partially wrong.

Let $0<a<1$ be arbitrary but fixed. The equation $$ \frac{x^2 (1+y)^2}{\left(a y + \sqrt{1+(1-a^2)y}\right)^2} = 1$$ in $y$ has according to straight-forward calculus and Mathematica two ...
2
votes
2answers
28 views

Continuity at a Point (Local Property)

Can anyone please explain me the following paragraph regarding limits and continuity?: One thing to note about continuity is that it is a local property. What this means is that, for any $a \in \...
2
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0answers
48 views

Prove that $12(a\sin a+\cos a-1)^2\le 2a^4+a^3\sin(2a)$,$\forall a\in (0,\infty)$

Prove that $$12(a\sin a+\cos a-1)^2\le 2a^4+a^3\sin2a,\forall a\in (0,\infty).$$ The solution in the book where I found this goes like this : from CS for integrals we have that $$\left(\int_0^a x\cos ...
2
votes
0answers
21 views

How to estimate $\sum_{2 \leq n \leq X} \frac{1}{(\log n)^A} $?

I was wondering how does one estimate $$ \sum_{2 \leq n \leq X} \frac{1}{(\log n)^A}? $$ where $A>0$. I feel like it should be $\ll \frac{X}{(\log X)^A}$... Any comments would be appreciated. ...
0
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1answer
27 views

Taylor-Remainder divided by $(x-x_0)^n$ goes to zero as $x$ approaches $x_0$.

In a script about the Taylor-Theorem I'm reading I have stumbled across the property that for remainder $R_n(x,x_0): \lim_{x \to x_0} \frac{R_n(x,x_0)}{(x-x_0)^n} = 0$. Since there are no further ...
2
votes
0answers
47 views

Let $f(x)$ is a stricly decreasing, continuous function from $[0, +\infty)$ to $[0, +\infty)$ such as $\lim_{x\to \infty} f(x)=0$ [duplicate]

Prove that $$\int_0^\infty {f(x)-f(x+1)\over f(x)}dx=\infty$$ I already understood that it is enough to prove that exist some constant $C$, such as for all $f$, exist $a_f$, such that $$\int_0^\infty ...
0
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1answer
22 views

Which of given condition follows from mean value theorem?

Let $f:R \to R$ be differentiable which of the following follows from mean value theorem : (a) For all $a,b \in R$ if $c \in (a,b)$, then $\dfrac{f(b) - f(a)}{b-a} = f'(c)$ (b) For some $a,b \in R$ ...
1
vote
1answer
41 views

Is my proof correct for $\tanh{(\tau+\epsilon)}=\tanh{(\tau)}+\epsilon(1-\tanh^2{\zeta})$ correct?

I am trying to prove an equality using Mean Value Theorem, I wish to know if my reasoning is correct, or if someone can improve upon the proof. The expression I am trying to prove is: \begin{equation} ...
9
votes
0answers
78 views

What exactly does curl measure? What is rotating about what?

This question is a bit long, the next two paragraphs give some context, but you can skip it. Thank you. I have seen many different explanations for the meaning of curl, or what exactly does it ...
1
vote
2answers
53 views

Convergence Theorems in the extended real numbers

I'm studying measure theory and usually functions take values in the extended real line. For most of the theorems, pointwise limits of such functions are considered. Now I was wondering whether some ...
4
votes
1answer
23 views

A property of oscillation function

I was solving an exercise from Stein-Shakarchi's Real Analysis, regarding the set of discontinuities of a Riemann integrable function $f$ on $[a,b]$. Let $\mbox{osc}(f,c,r)=\sup \{ |f(x)-f(y)| : x,...