Are step functions used in double integral definition bounded?
Definition of double integral on a rectangle can be defined by using step functions, for example in Apostol Calculus Book. In this case the definition for step function is: * a function $s$ defined...
View ArticleMust the fixed point of a function g commuting with a contraction be unique?
Let $(X,d)$ be a complete metric space. Suppose $f: X\to X$ is a contraction and $g: X\to X$ is a function commuting with $f$, i.e.,$$ f(g(x)) = g(f(x)),\quad \forall x\in X.$$Clearly, $g$ has at least...
View ArticleFinding a suitable $\delta$ for $\varepsilon$-$\delta$ proofs for third...
In class my professor went through an $\varepsilon$-$\delta$ proof of the limit $$\lim_{x\to 4} (x^3 -8x^2 + 16x + 4) = 4.$$He found a suitable $\delta = \min \left\{1, \sqrt{\frac{\varepsilon}{5}}...
View ArticleQuestion about Riemann integrability: do we need to specify that all Riemann...
Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. Suppose that there is a sequence of partitions $\{P_n\}_{n=1}^\infty$ with mesh tending to $0$, $P_n=\{a=t_0^n<t_1^n<\ldots<t_{r_n}^n=b\}$,...
View ArticleProof That There is a Sequence Converging to $f'(c)$ [duplicate]
Suppose $f:[a,b]\to\mathbb{R}$ is differentiable and $c\in{[a,b]}$. Prove that there exists a sequence $\{x_n\}_{n=0}^{\infty}$ such that ${\forall{n}\in\mathbb{N}}, x_n\neq{c},...
View ArticleWhy is the value of the series representation for $\arctan$ at $ x = 1$...
Exercise 6.6.1. The derivation in Example 6.6.1 shows the Taylor series for $\arctan(x)$ is valid for all $x \in (−1,1)$. Notice, however, that the series also converges when $x = 1$. Assuming that...
View Articlefixed accumulation point weak probability measure on subsets of reals.
I am working on a joke real analysis course with some friends and I have a problem which I managed to reduce to the existence of a function $P:2^{\mathbb{R}}\to\mathbb{R}$ for a fixed $a\in\mathbb{R}$...
View ArticleLebesgue integral of $L^1$ function is differentiable
Let $f\in L^1(\mathbb{R})$, and define the function$$F(x)=\int_a^xf(t)dt.$$I want to prove that $F$ is almost everywhere differentiable and that $F'(x)=f(x)$ where $F$ is differentiable.I am following...
View ArticleIs my derivation of Taylor's Formula in $\mathbb R$ for the remainder...
Statement from a textbook:Let $n \in \mathbb N$ and let $a,b$ be extended real numbers with $a<b$. If $f:(a,b)\rightarrow \mathbb{R},$ and if $f^{(n+1)}$ exists on $(a,b)$, then for each pair of...
View ArticleShow that $\forall x,y \in (a,b):x
Let $a<b, (a,b \in \mathbb{R})$I want to show that there is an homeomorphism $f$ between $(a,b)$ and $\mathbb{R}$ such that $x < y \iff f(x) < f(y)$, $(x,y \in (a,b))$, but I don't want to use...
View ArticleTotally bounded set in a metric space $\implies$ bounded
I apologize if the question may be trivial, but it is a fact that my textbook does not even mention and I, studying as a self-taught, do not have so many certainties.I believe the totally boundedness $...
View ArticleAn inequality related to BMO on [0,1]
Suppose that $b\in L^1[0,1]$, and that $I$ is a subinterval of $[0,1]$ centered at $s_0$. we denote by $\widetilde{I}$ the double of $I$, namely, the interval centered at $s_0$ with length $2|I|$.I...
View ArticleIf we only specify one sequence of partitions in the definition of...
Update: This question differs from Question about Riemann integrability: do we need to specify that all Riemann sums converge to the same number in the definition? in the following way:From what I see...
View ArticleSecond Countability of Euclidean Spaces
Sorry I know this is a stupid question. However I got stuck on this for quite a while. I'm trying to prove that Euclidean spaces have a countable base, which can be constructed by taking all the open...
View ArticleDiscuss the spectrum of the integral operator $Tu(x)=\int_0^1 g(x,t)u(t)dt$...
Given $g\in C^1([0,1]\times[0,1])$, consider the operator $$Tu(x) = \int_0^1 g(x,t) u(t) dt$$ defined on $u\in C([0,1])$. Discuss the spectrum of T.My attempt:First I can show that $T$ is a compact...
View ArticleInequality on Sobolev norm
I have a problem with the following inequality. Suppose r and r' are Holder conjugate, i.e. $\frac{1}{r} + \frac{1}{r'} = 1$. Then we have$$|||u|^{\alpha -1} u||_{W^{1, r'}(\mathbb{R}^n )} \leq c...
View ArticleDoes there exist an ultrafilter of $\mathbb{R}$ with fixed accumulation point...
This question arose from a real analysis question, where i am more generally looking for a finitely additive measure on $2^{\mathbb{R}}$ for a fixed point $a$ that satisfies the property:$\forall A \in...
View ArticleDoes the integral $\int_{0}^{\infty} \frac{x}{1 + x^5 \sin^2(x)} \, dx$...
$$\int_{0}^{\infty} \frac{x}{1 + x^5 \sin^2(x)} \, dx$$If you could provide a detailed solution process, I would be extremely grateful!
View ArticleSuppose $x_n < 0$ for all $n$. If $\lim(x_n) = 0$, prove that $\lim(1/x_n) =...
How do you go by proving this one . any hint ?I have no idea where to start with this one
View ArticleExistence of global minimum of convex function
Let $f:I\longrightarrow \mathbb{R}$ a convex function cotinous on a interval $I$.If $f$ not is monotone, then is true that $f$ has a global minimum in inner of $I$?--My attempt: because $f$ is no...
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