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$$g_n : X_0...
View ArticleContinuity + Banach (S) property implies bounded variation?
Are the following implications true:(I) if $g$ is continuous on $[a,b]$, and if $g$ satisfies the Banach (S) property on $[a,b]$, then $g$ has bounded variation on $[a,b]$? Here we assume that $a,b\in...
View ArticleHow to prove that function doesn’t have primitive function? [closed]
How to prove that f doesn’t have primitive function?$$f(x) = \begin{cases}\sin^{2}(\frac{1}{x}),& x \neq 0 \\\ 0, &x = 0 \end{cases}.$$I did not found nowhere criterias for function that...
View ArticleLet $\{ a_n \}_n$ be a sequence such that $a_n < \frac{a_{n-1} +...
Let $\{a_n \}_n$ be a sequence such that $a_n \geq 0 \, \forall n \in \mathbb{N}$ and$$a_n < \frac{a_{n-1} + a_{n-2}}{2}$$Prove that $\{ a_n \}_n$ is convergentMy attempt: I tried to prove that the...
View ArticleUsing the derivative, find $a$ such that: $|x-a| = (x-2)^2$
So I've been studying Jay Cummings's Real Analysis book, and I've encountered this problem:Use the derivative to find all values of $ a $ such that the following holds: $$|x-a| = (x-2)^2 $$He doesn't...
View ArticleNeed help solving $\lim_{x\to \infty}\frac{\sqrt x}{\cos(2x)+6x}$ and...
For the first I tried making it $\frac{\frac{1}{\cos(2x)+6x}}{\frac{1}{\sqrt x}}$ so it can be $\frac{0}{0}$ and do de l'hospital but it goes again at $\frac{0}{0}$ so I suppose I just dont know...
View ArticleFinding supremum of an absolute power series.
Find $\lambda > 0$ and $0 < \mu < \frac {1} {2}$ such that $$\sup\limits_{0 \leq r < 1} \sum\limits_{n = 0}^{\infty} \frac {\Gamma^2 \left (\frac {3 \lambda} {2} + n \right )} {(n!)^2}...
View ArticleQuestion 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...
View ArticleIntermediate 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,...
View ArticleShow that $f_k$ has no continuous limit function
Let be $f_n:[0,1]\to\mathbb{R}$ where $f_n(x):=\begin{cases}0,&x\in...
View ArticleAn exercise on convex decreasing function properties
A function f$(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...
View ArticleProving 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...
View ArticleStieltjes 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 $g(x) = x - \operatorname{sgn}(x)R$....
View ArticleFolland 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...
View ArticlePreserving 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...
View ArticleFind 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...
View Articlewhat 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...
View ArticleRelation 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}...
View ArticleRadius 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...
View ArticleCan 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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