Hello, please I need help with exercise 21. I can't get it done. Thank you...
[Hello, please I need help with exercise 21. I can't get it done. Thank you.][1]![real analysis]: https://i.sstatic.net/o1jB8hA4.jpg
View ArticleWhen can we write $\lim_{n\to\infty}\sum_{j=1}^\infty e^{ij}f(\frac...
Let $f_2$ be an integrable function. I am trying to sum$$\lim_{n\rightarrow\infty}\sum_{j=1}^{\infty}e^{ij}f_{2}\left(\frac{j}{n}\right)\frac{1}{n}$$Under what condition can I write this...
View ArticlePlease explain the second paragraph of the following image.
Taylor's Theorem's Converse.Please help me understand the validity of Taylor Theorem's converse.
View Articleupper and lower sums in Riemann integral
I want to prove this:If $P^{*}$ is a finer partition than $P$, then show that $L(f,P, \alpha) \leq L(f,P^{*}, \alpha)$ and $U(f,P^{*}, \alpha) \leq U(f,P, \alpha)$.If you have a set $S = \{1.2.3 \}$...
View ArticleRiemann stieltjes integration lower sum rectangle area
If P∗ is a finer partition than P, then show that L(f,P,α)≤L(f,P∗,α).explain me how area of rectangle of lower sum is L(p,f,α) = inf f(x). [α(b)-α(a)] and not [inf f(x)- inf α(x)][b-a]and apart from...
View ArticleComposition of a function with a metric
Which of the following functions, $f: [0,\infty) \rightarrow [0,\infty)$, can be composed with a metric $d$ to get a new metric $f \circ d$: a)$\;f(x) = \begin{cases}0 & \text{if $x=0$} \\x+1 &...
View ArticleSum of continuous functions is continuous with multiple variables
BackgroundI have seen proofs showing that the sums of two continuous functions $f_1, g_1 :$$\mathbb{R} \rightarrow \mathbb{R} $ are continuous, and I have also seen this result for functions $f_2, g_2...
View ArticlePower of Cosine Inequality
The problem is such:Show that $$\cos{px}\ge \cos^p{x}$$ given that $0\le x\le \pi/2$ and $0<p<1$I'm pretty sure this is straight forward given the sum formula for...
View ArticleProving that a curve is $C^\infty$
Let $\alpha : [0, + \infty) \rightarrow \Bbb{R}^2$ be a map defined as $\alpha(0) = (0,0)$ and $\forall t > 0 : \alpha (t) = (e^{-1/t} \cos (1/t) , e^{-1/t} \sin (1/t)) $ . Prove that $\alpha$ is...
View ArticleAsk the proof for bounded and finite-dimensional implies totally bounded?
The definition of totally bounded is: A metric space $(\mathbb{M}, d)$ is totally bounded if for any $\epsilon>0$ there exists $y_1, \ldots, y_n$ in $\mathbb{M}$ such that $\mathbb{M} \subset...
View ArticleDoes $y\in[0,\:1]\implies y\in f((-1,\:1])$?
Suppose that $f(x)=x^2$. Find $f((-1,\:1])$.$y\in f((-1,\:1])$ iff $y=f(x)$ for some $x\in(-1\:1]$iff $y=x^2$ for some $x\in(-1\:1]$iff $y=x^2$ for some $x$ such that $-1<x≤1$iff $y=x^2$ for some...
View ArticleDoes a function that is differentiable at a point satisfy the Lipschitz...
If I have a function $f(x)$ defined on a compact interval, which is differentiable at the point $x=x_0$, does it follow that it satisfies the Lipschitz condition around that point? As in, is it true...
View ArticleIs this convex set open? [closed]
Let $X$ be a normed space over $\mathbb{R}$, and let $K \subset X$ be a convex subset with the property that, for every $u \in X, u \neq 0$ there exists $M(u) > 0$ such that$ \{ \lambda \in...
View ArticleRepresent a line as an equality with 0
I have the following line in $\mathbb{R}^3$:$$r\equiv x=y=z$$I can also represent $r$ with a set of equalities like:$$r\equiv\begin{cases}2x = y + z \\2y = x + z \\2z = x + y\end{cases}$$However, I...
View ArticleLocally Lipschitz function and linear growth [closed]
I know that if $g: \mathbb{R}^d \to \mathbb{R}$ is Lipschitz, this means that there exists $L > 0$ such that$$|g(x) - g(y)| \leq L |x - y|,$$for all $x, y \in \mathbb{R}^d$. Therefore, by taking $y...
View Articlef is a homeomorphism thrn show that the following statement are equivalent...
Let $f : (M, d) \to (N, p)$ be one-to-one and onto. Then the following are equivalent:$f$ is a homeomorphism.$d_1(x , y) = p(f(x), f(y))$ defines a metric on $M$ equivalent to $d$.
View ArticleA smooth extension with an assigned regular value.
Let $f\colon S^{n-1} \to \Bbb R^n$ be a smooth function from the unit sphere into $\Bbb R^n$ (not necessarily injective) and $y_0 \in \Bbb R^n\backslash f(S^{n-1})$ be a given point. Can we always...
View ArticleSome properties of the averaged modulus of smoothness
I am currently studying a paper published in Journal of approximation theory and trying prove unsolved properties.Following are the definitions required in properties and properties itself:The averaged...
View ArticleCondition for existence of Fourier transform?
We can convert signal into frequency domain using Fourier transform. But I think we can't compute Fourier transform of any signal . Fourier transform also should have some limits.So I want to askis...
View ArticleMeasurable set - A sequence of measurable functions
Let $(X, M, \mu)$ be a measure space, and $f_n:X \rightarrow \mathbb R$ sequence of measurable functions.How can I show that the set of $x$ that for them $f_n(x)$ has a subsequence that converges to...
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