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Is an infinite composition of bijections always a bijection?

Main QuestionSuppose I have a sequence of real valued functions $f_1:X_0\rightarrow X_1,...,f_n:X_{n-1} \rightarrow X_n,...,$ and I then, with $\circ$ denoting function composition, define$$g_n : X_0...

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Continuity + Banach (S) property implies bounded variation?

Are the following implications true:(I) if $g$ is continuous on $[a,b]$, and if $g$ satisfies the Banach (S) property on $[a,b]$, then $g$ has bounded variation on $[a,b]$? Here we assume that $a,b\in...

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How to prove that function doesn’t have primitive function? [closed]

How to prove that f doesn’t have primitive function?$$f(x) = \begin{cases}\sin^{2}(\frac{1}{x}),& x \neq 0 \\\ 0, &x = 0 \end{cases}.$$I did not found nowhere criterias for function that...

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Let $\{ a_n \}_n$ be a sequence such that $a_n < \frac{a_{n-1} +...

Let $\{a_n \}_n$ be a sequence such that $a_n \geq 0 \, \forall n \in \mathbb{N}$ and$$a_n < \frac{a_{n-1} + a_{n-2}}{2}$$Prove that $\{ a_n \}_n$ is convergentMy attempt: I tried to prove that the...

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Using the derivative, find $a$ such that: $|x-a| = (x-2)^2$

So I've been studying Jay Cummings's Real Analysis book, and I've encountered this problem:Use the derivative to find all values of $ a $ such that the following holds: $$|x-a| = (x-2)^2 $$He doesn't...

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Need help solving $\lim_{x\to \infty}\frac{\sqrt x}{\cos(2x)+6x}$ and...

For the first I tried making it $\frac{\frac{1}{\cos(2x)+6x}}{\frac{1}{\sqrt x}}$ so it can be $\frac{0}{0}$ and do de l'hospital but it goes again at $\frac{0}{0}$ so I suppose I just dont know...

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Finding supremum of an absolute power series.

Find $\lambda > 0$ and $0 < \mu < \frac {1} {2}$ such that $$\sup\limits_{0 \leq r < 1} \sum\limits_{n = 0}^{\infty} \frac {\Gamma^2 \left (\frac {3 \lambda} {2} + n \right )} {(n!)^2}...

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Question About Dense Subspace of $L^p(X,\mathscr{A},\mu)$ - Proof of Theorem...

BackgroundI am self-studying Donald Cohn's Measure Theory second edition. I got stuck on a step of the proof of Theorem 4.5.1. Here is the theorem:Theorem$\quad$Let $(X,\mathscr{A},\mu)$ be a measure...

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Intermediate Value Theorem, Proof review

I wonder if this proof is alright and I can take it into my notebook. It is my version of the standard version, so that it makes more sense to me. If there are significant mistakes or shortcomings,...

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Show that $f_k$ has no continuous limit function

Let be $f_n:[0,1]\to\mathbb{R}$ where $f_n(x):=\begin{cases}0,&x\in...

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An exercise on convex decreasing function properties

A function f$(x)$ defined for $x\geq0$. It is positive, decreasing, convex and log-convex: $\frac{d^2}{dx^2}\log[f(x)]>0$, $f(0)<1$. Can we prove that $f''(x)x+f'(x)>0$ for sufficiently large...

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Proving sequences are Cauchy in a metric space

In this question I have concluded that the sequences in b) and c) are both Cauchy with respect to $d$. This is because I deduced that $f(a_1 + \ldots + _n) \geq min\{f(a_1), \ldots, f(a_n)\}$ and used...

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Stieltjes Integration Problem

Consider a non-negative bounded function $f:[-R,R]\rightarrow\mathbb R$ with infimum bounded away from $0$, and the function $g:[-R,R]\rightarrow\mathbb R$ with $g(x) = x - \operatorname{sgn}(x)R$....

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Folland Question 6.4 trouble with estimating norm

The question is as follows:If $1\leq p<r\leq \infty$, prove that $L^p+L^r$ is a Banach space with norm $\lVert f\rVert= \inf\{\lVert g\rVert_p+\lVert h\rVert_r\,|\, f=g+h\in L^p+L^r\}$, and prove...

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Preserving convergences under composition of function sequences

I have following propositions to decide whether they are true or false(Also checked some problems in site such as Question1 ,Question2)Let $g:A\subset \mathbb{R} \to \mathbb{R}$ bijection and for all...

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Find all Real Numbers such that $\lfloor x\lfloor x\rfloor\rfloor=2024$

I just came across an Olympiad problem that says:$$\lfloor x\lfloor x\rfloor\rfloor=2024$$I have never encountered floor functions in my life and so I have tried this approach to...

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what is $\lim_{x\to 0} f(x)$

Let $$f(x)=\sum_{n=1}^{\infty}{\sin(nx)\over n^2}$$ Then what is $\lim_{x\to 0} f(x)$Now I know the series converges uniformly by $M-Test$ (Take $M_n=1/n^2$). What should be my next step. I am...

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Relation between supremum and derivative of integral

Several times I come across the following transition between the quantity for some bounded Euclidean subset $U$$$\int_0^T \int_{U} \dfrac{\partial}{\partial t} f(x,t)dx dt$$to$$\sup_{0<t<T}...

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Radius of convergence of real analytic functions

Let $f: \mathbb{R} \to \mathbb{R}$ be a function that is analytic at every point in $\mathbb{R}$ except at a single point $x_0$.Radius of Convergence Near $x_0$:Can we prove that the radius of...

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Can a function with just one point in its domain be continuous?

For example if my function is $f:\{1\}\longrightarrow \mathbb{R}$ such that $f(1)=1$.I have the next context:1) According to the definition given in Spivak's book and also in wikipedia, since...

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