Existence of $\lim_{x→0} f(x \sin(1/x))$ implies continuity of $f$ at $0$?...
Let $f: \Bbb{R}\to\Bbb{R}$. Assume that the limit $\lim_{x→0} f(x \sin \frac 1 x)$ exists. Must $f$ be continuous at $0$?Using the sequence $x_n = \frac 1 {\pi n}$ we must have that $\lim_{x→0} f(x...
View ArticleMollification and limit
Consider the $d$-dimensional torus $\mathbb{T}^d.$ Let $\phi \in C_c^\infty(\mathbb{R}^d,\mathbb{R})$ such that $\phi(0)=1$ and $\phi_\epsilon(x):=\phi(\epsilon x),\rho\in C^\infty(\mathbb{T}^d).$Prove...
View ArticleShow that there is $E \in M$ such that $\mu(E)=\alpha$ for any $0 \lt \alpha...
Let $(X,M,\mu)$ be a finite measure space with no atoms. A set $A \in M$ is called an atom if $\mu(A) \gt 0$ and for any measurable subset $B \subset A$ ,either $\mu(B)=0$ or $\mu(A-B)=0$ .Show that...
View ArticleThe upper bound of Lipschitz function with cell problem
Let us consider the Hamiltonian $H = H(y,p): \mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$,which is continuous and satisfies below.(H1) For each $p\in\mathbb{R}^n, y\mapsto H(y,p)$ is...
View ArticleIs there a closed form for the linear operator $T$ such that $T(x^n)...
When I first learnt calculus I was so surprised to learn that there is a meaningful mathematical operator $D$ that$$D(x^n)= n x^{n-1}.$$It seemed to be a very random thing to multiply the exponent by...
View ArticleProving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$...
Here is an exercise, on analysis which i am stuck. How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on $\mathbb{R}$?
View ArticleIf a function is integrable and is of bounded variation, then must...
Let $f:(0,\infty)\to [0,\infty)$ be a continuously differentiable function.Assume that $\int_0^{\infty} f(x)dx<\infty$ and that $\int_0^{\infty}|f'(x)|dx<\infty$.Claim: $\sum_{n=1}^{\infty}f(n)$...
View ArticleA question related to physical application of differentiable functions in...
A body that can be regarded as a point mass is sliding down a smooth hill under the influence of gravity. The hill is the graph of a differentiable function $y = f (x)$.a) Find the horizontal and...
View ArticleHow to show that the inf / sup of the set of the $n$-th powers of the...
Let $A$ be a nonempty, bounded (from above) set consisting of non-negative real numbers only, let $n$ be a given positive integer, and let sets $B$ and $C$ be defined as follows:$$B := \left\{ a^n...
View ArticleIf f(x) and g(x) have equal value and the same derivative at x=0, do they...
In Wikipedia page on germ, it says thatGiven a point $x$ of a topological space $X$, and two maps $f,g\,:\,X\to Y$ (where $Y$ is any set), then $f$ and $g$ define the same germ at $x$ if there is a...
View ArticleEvaluating $\lim_{x\to \infty} \frac{f^{-1}(2021x)-f^{-1}(x)}{\sqrt[2021]...
Let $f(x)=2021x^{2021}+x+1$, and compute the following limit:$$\lim_{x\to \infty} \frac{f^{-1}(2021x)-f^{-1}(x)}{\sqrt[2021] x}$$My attempt: i want to use mean value theorem to $f^{-1}(x)$ then we...
View ArticleEquidistribution Theorem for Vectors
I have seen the following question, where the equidistribution theorem can be generalized to 2 dimensions under certain conditions: Equidistribution in higher dimensions.I was wondering if the same...
View Article$\lim_{n\to\infty}\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}$
I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}$$ where $\{x\}$ denotes the fractional part of $x$ and $n\in\mathbb{N}$.By definition of...
View ArticleExistence of degree $n$ best approximation polynomial implies Existence of...
There exist a degree $n+1$ polynomial of best approximation if there exist a degree $n$ polynomial of best approximationThis is the context of the question.If there exists a polynomial of best...
View ArticleT- separating of continuouse function space that are zero in infinite on...
$X$ is compact hausdroff topologic space.$\tilde{X}=\{(x,y)\in X×X ; \hspace{2mm} x\neq y\}$.Could you help me in proving that $C_0(\tilde{X})$ is a $\mathbb{T}$-separator?
View ArticleEquivalence of Logarithm Definitions
As discussed in this question, there are many different approaches to defining the natural logarithm function. In particular, since the exponential function$$\exp(x) :=...
View ArticleProve that $\int_0^1 \left| \frac{f^{''}(x)}{f(x)} \right| dx \geq4$.
Let $f(x) \in C^2[0,1]$, with $f(0)=f(1)=0$, and $f(x)\neq 0$ when $x \in(0,1)$. Show that$$\int_0^1 \left| \frac{f''(x)}{f(x)} \right| dx \geq 4.$$
View ArticleCan there exist a function $F$ which is differentiable on $[a,b]$ but $F'$ is...
QuestionCan there exist a function $F$ which is differentiable on $[a,b]$ but $F'$ is non-riemann integrable function on $[a,b]$. There is a similar question here, but the construction done is not...
View ArticleHow to prove that $ \underset{\varDelta x\rightarrow 0}{\lim}\frac{\left| MN...
In the derivation of the arc differential formula, why is the limit of the ratio of the length of the line segment between two points to the length of the arc considered to be 1: $$\underset{\varDelta...
View ArticleOn a notion of asymptotically monotone sequence
Let us say that an infinite sequence $\{a_n\}$ of real numbers is asymptotically increasing if there exists a double sequence$\{b_{n,m}\}_{n<m}$ such that(1) $b_{n,m}\rightarrow 0$ as $n,m...
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