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.

Filter by
Sorted by
Tagged with
7
votes
2answers
149 views

Interesting relation derived from the identity $\sin^2 x + \cos^2 x = 1$

Every one knows that $$\sin^2 x + \cos^2 x = 1.$$ It is also well known that $$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n + 1}}{(2n+1)!},\quad\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(...
1
vote
0answers
12 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
votes
1answer
17 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}{...
25
votes
5answers
29k views

$\sqrt x$ is uniformly continuous

Prove that the function $\sqrt x$ is uniformly continuous on $\{x\in \mathbb{R} | x \ge 0\}$. To show uniformly continuity I must show for a given $\epsilon > 0$ there exists a $\delta>0$ such ...
-4
votes
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 ...
0
votes
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
votes
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 ...
0
votes
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
votes
0answers
64 views
+50

Sequence problem regarding convergence from an online contest

Let $(x_n)_{n\in \mathbb{N}}$ be a sequence defined by $x_0=1$ and $x_n=x_{n-1}\cdot (1-\frac{1}{4n^2})$,$\forall n\geq 1$. Prove that : a)$(x_n)_{n\in \mathbb{N}}$ is convergent b) if $l=\lim_{n\to \...
1
vote
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 ...
1
vote
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,...
1
vote
1answer
19 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 ...
4
votes
0answers
277 views

How can one show the multiple integral $\int_{0}^{a} \int_{0}^{a} \frac{dx ~ dy}{(a^2+x^2+y^2)^{3/2}}= \frac{\pi}{6a}$ by hand? [duplicate]

Mathematica gives the value of the multiple integral $$ \int_{0}^{a} \int_{0}^{a} \frac{dx ~ dy}{(a^2+x^2+y^2)^{3/2}}=\frac{\pi}{6a}. $$ See also this result from WolframAlpha: The question is ...
-6
votes
0answers
29 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 ...
0
votes
0answers
21 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.
13
votes
3answers
1k views

Methods to evaluate $ \int _{a }^{b }\!{\frac {\ln \left( tx + u \right) }{m{x}^{2}+nx +p}}{dx} $

Today I saw a question with an answer that made me rethink of the following question, since it's not the first time I try to find an answer to it. If you look at the answer of Mhenni Benghorbal here ...
-2
votes
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
votes
0answers
42 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
votes
1answer
115 views

Can we define a metric on R such that all sequences are convergent?

Can we define a metric on $\mathbb{R}$ such that all sequences are convergent?
3
votes
1answer
52 views

Can we approximate elements in $L^2$ via smooth maps while preserving a pointwise constraint on the derivatives?

Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball. Suppose we are given an open connected* subset $U \subseteq \mathbb{D}^n$ of full measure in $\mathbb{D}^n$, and a ...
1
vote
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^{...
2
votes
1answer
51 views

$\lim_{n\to \infty} \frac{(-1)^{\frac{(n-1)(n-2)}{2} }}{n} $, solving this limit using Stolz theorem and getting the wrong result

$$\lim_{n\to \infty} \frac{(-1)^{\frac{(n-1)(n-2)}{2} }}{n} $$ It is easy to see that this is zero since the demoinator goes to infinity and the numerator oscillates between -1 and 1.But if i try to ...
0
votes
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 ...
3
votes
3answers
41 views

How to check if a function is convex

According to a calculus book I have been reading, we call a function $g(x)$ a convex function if $$g(\lambda x +(1-\lambda)y) \leq \lambda g(x) +(1-\lambda)g(y)$$, for all $x,y$ and $0<\lambda&...
1
vote
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 \}$. ...
3
votes
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 ...
5
votes
3answers
2k views

Are undefined terms allowed in a sequence?

Is this a valid sequence: $\{\frac1{(n-3)}\}$? I.e., can a sequence have individual terms that are undefined? And if so, does this mean that the above sequence is unbounded (since the third term is ...
-1
votes
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 $\lambda_{2}$ $$ \...
6
votes
3answers
147 views

Is the space of maps which satisfy this vanishing condition finite-dimensional?

Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball. Let $h:\mathbb{D}^n \to \mathbb{R}^{k}$ be smooth, and suppose that $h(x) \neq 0$ a.e. on $\mathbb{D}^n$. Set $$V_h=\...
1
vote
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 ...
6
votes
2answers
86 views

Is there accepted notation for specifying just the domain of a function?

Question. Is there a notation like $$f(x \in \mathbb{R}) = x^2 + 2x + 1$$ or some variant on that, satisfying the following conditions? (a) Like the above syntax, it allows us to define a ...
1
vote
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 ...
1
vote
1answer
469 views

Intersection of open dense subsets $G_n$ of $R^k$ n = 1,2… is not empty (more over its dense)

I am attempting to prove a special case of Baire theorem (ex. 30 in Rudin book). It states: If $G_n$ is a dense open subset of $\mathbb{R}^k$ for n = 1,2,3..., then $\bigcap_nG_n\neq \varnothing$ (in ...
2
votes
4answers
165 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 ...
2
votes
1answer
572 views

If one of the Dini derivatives is bounded, then f is Lipschitz

If one of its Dini derivatives (say $D^+$) is bounded show that a function $f$ satisfies a Lipschitz condition, Definition of the upper right Dini derivative: $$D^{+}f(x) = \limsup_{h\to\ 0^+} \...
24
votes
5answers
2k views

What's the “limit” in the definition of Riemann integrals?

Consider one of the standard methods used for defining the Riemann integrals: Suppose $\sigma$ denotes any subdivision $a=x_0<x_1<x_2\cdots<x_{n-1}<x_n=b$, and let $x_{i-1}\leq \xi_i\...
2
votes
0answers
51 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
vote
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
votes
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 ...
2
votes
2answers
68 views

Essence of Weierstrass approximation theorem.

Weierstrass approximation theorem is a quite strong theorem,even stronger than the Taylor's theorem because: Statement:Suppose $f:[a,b]\to \mathbb R$ is a continuous function then $\exists$ a ...
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
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 ...
4
votes
1answer
70 views

Proving the existence of a continuous function with same integral as another function

Let f be a Riemann-integrable function in $[a,b]$. I have to prove that $\forall \epsilon >0 , \exists g$ continuous , such that $ g \leq f$ and $\int_a^b f - \int_a^b g < \epsilon $. I thought ...
1
vote
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 ...
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
votes
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
3answers
1k views

Let X be a metric space in which every infinite subset has a limit point. Prove that X is compact.

Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is compact. The following is my proof I'd like to know if it is correct. Proof: I will use the fact that ...
0
votes
0answers
38 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 ...
0
votes
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
vote
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_{...