$C^n(X)$ separability
Using Stone-Weiestrass thm we can say easily that $(C^n(K), \| . \|_{\infty})$ where $K \subset \mathbb{R}^n$ compact, is separable as subspace of separable metric space.Anyway i'm interested to the...
View ArticleEvaluate $\sum\limits_{n=1}^\infty...
I am trying to find the pattern for the coefficients of the closed forms for this series:$$\sum_{n=1}^\infty \left( \frac{1}{(k+n)^2+n^2}-\frac{1}{(k+1-n)^2+n^2} \right)$$The series seems to converge...
View ArticleTotal variation of integral function
Consider a function $f\in L^1([a,b])$ and define $F(x):=\int_a^x f(y)dy$. I should prove that $V_a^bF=||f||_{L^1([a,b])}$.My attempt was to proceed by approximation with a test function $\phi$, but I’m...
View ArticleExistence of everywhere continuous, nowhere differentiable function with...
Does there exists a function $f:[0,1]\to[0,1]$ such that:$(1)$$f$ is continuous everywhere.$(2)$$f$ is differentiable nowhere. $(3)$ For every $c \in range(f)$, there is only a finite number of points...
View Articleis the equation has root in $(0,1)$?
for $a,b,c \in \mathbb{R}$, the equation$2cx^5+(4c+3b)x^4+(2c+6b)x^3+(3b+a)x^2+2ax=a+2b+c$has root in $(0,1)$?I tried to use IVT but I can't this problem.Does anyone know if this it true and how to...
View ArticleDensity Theorem and Limit Proof
I'm new to proof writing and have trouble understanding hoe to structure or get proofs started. An example proof that my teacher left as practice is below. Any suggestions? Of course, I know I must use...
View ArticleI have a one question. What i need to do to translate function f(x,y,z) to...
look f(x,y,z)=x^4+6y^8+17z^9 and what to do to make something with z to get f(x,Y) but do not give 0 to z,x,y x,y,z∈R R= real number
View ArticleFinding functions such that the maximum is in certain values
Let $a_1, a_2, b_1, b_2, r_1, r_2 \geq 0$ with $a_1 > b_1$ and $b_2 > a_2$ be pre-determined constants. Define the function $h$ as $$h(x, y) := f(x, y) (a_1 r_1 + a_2 r_2) + g(x, y) (b_1 r_1 +...
View ArticleDoes this sequence of function converges uniformly?
I was practicing for my final and stumbled on this problem.Let $f_n:[0,1]\longrightarrow\Bbb R$ such that$$f_n(x)=\begin{cases}nx, & 0\leq \text{ $x$ } \lt1/n\\1, & 1/n\leq\text{ $x$ }\leq...
View ArticleFind minima of this function
I want to know if my solution for this problem is correct:Let $f : [0,\infty ) \to [0,\infty)$ with continous first derivative. Suppose that for all $a,b \in \Bbb{R^+}$, the area between the $a,b$ is...
View ArticleUniform convergence of $f_n(x) = \frac{\ln(1+\frac{x}{n})}{x+1}$ [duplicate]
Basically i have a problem proving the sequence in the title 1. Uniformly converges on a closed interval $[0,a]$ where $a > 0$ and 2. Uniformly converges on $[0,\infty).$ So far i have found the...
View ArticleName of a function that can not be described by a lower-dimensional function
I am looking for a good name of a function $f:\mathbb{R}^n\to \mathbb{R}$ satisfying the following property:There does not exist function $g:\mathbb{R}^k\to \mathbb{R}$, $k<n$, such that...
View ArticleCan $\sin (x)$ be considered zero at the point at $\infty$ in some context?
It is understood that $\lim_{x \to\infty} \sin (x)$ is not defined, but I have tricked myself into thinking it might be identifiable as zero at the point at infinity after thinking about continuity,...
View ArticleFunction $\frac{x}{1+x}g(a-x) + \frac{1}{1+x}g(-x)$ has a unique maximizer...
Let $g:(-L,\infty) \to \mathbb{R}$ be a strictly concave, increasing, and differentiable function with $L > 0$ such that $g(x)\to -\infty$ as $x \to -L$. Also, let the map $f$ be defined on $[0,L)$...
View ArticleIssue with sum advanced sum.
my friend has sent me this series problem$\displaystyle\sum_{n=1}^\infty \frac{1\cdot4\cdot7\cdots(3n-2))}{5^n \cdot n!}$And he showed me that it converges to$(\frac{5}{2})^{1/3} -1$Wich...
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Minimize $\frac{1}{2}\sum_{k=1}^m (x_{k+1}-x_{k})^2$
Given sequence:$$\begin{cases}x_{n+1}(2\cos(\frac{\pi}{m})-x_n)=1,\forall n\geq 1\\x_1=x\in\mathbb R,m\in\mathbb N,m\geq 2\end{cases}$$Minimize $$A=\frac{1}{2}\sum_{k=1}^m (x_{k+1}-x_{k})^2$$EditI've...
View ArticleHow pathological can an unbounded function with a closed graph be?
This question has been inspired by the answers and comments received on this sister question.Let $I\subseteq\mathbb R$ be a non-degenerate closed (but not necessarily bounded) interval. Suppose that...
View ArticleHow to prove that outer measure $|A|=\lim_{t\to \infty}|A\cap (-t,t)|$?
Let $A\subset R$. Then, $|A|=\lim_{t\to \infty}|A\cap (-t,t)|\tag 1$$|A|:=\inf\{\sum_j l(I_j): A\subset \cup I_j\},$ where $I_j$'s are open intervals in $\mathbb R$ and $l(I_j)$ is the length of the...
View ArticleIs $\int_D \nabla f dx = 0$ for $f$ compactly supported?
Let $D \subset \mathbb{R}^n$ be bounded and $f$ a smooth compactly supported function such that its support is contained within $D$. I am interested in$$\int_D \nabla f dx.$$If $n = 1$, then by the...
View ArticleDo rational functions eventually have monotonic derivatives?
Given a rational function $R(x)=P(x)/Q(x)$ with real coefficients, is it true that there exists an $M>0$ such that, for every $k\geq 0$, the restrictions $R^{(k)}|_{(-\infty,-M]}$ and...
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