Uniformly bound exists for a continuous function sequence in a neighborhood...
Assume that $$\lim_{n\to\infty}f_n(x_0) < \infty$$ and also that $\forall n\ge1$, $f_n(x)$ is continuous in a neighborhood of $x_0$, say $(x_0-\delta_1, x_0+\delta_1)$. Besides, for any fixed $x$ in...
View ArticleBound on derivatives of compactly supported function
Let $f$ be a smooth function of compact support on $\mathbb R$.Let$$a_n=\sup_{x\in\mathbb R}|f^{(n)}(x)|.$$I would be interested in upper bounds of the growth of this sequence. In particular: does...
View ArticleMeasure theory, prove the countable additivity of measure
I am reading Cohn's Measure Theory, here is an exercise from Chapter 1, Section 2.Let ($X,\mathcal A,\mu$) be a measure space, and define $\mu^{\star}:\mathcal A\rightarrow[0,\infty)$...
View ArticleEvaluating a limit given the limit of the derivative as $x$ approaches...
I have been asked to evaluate the following limit as a challenge problem in my real analysis class. This was part of a problem set which mainly involved L’Hopital’s rule, but since you can’t use it...
View Articleequality of inner product and derivative
I came across the following claim$$\lim_{r \to 0} \dfrac{f(rx) - f(0)}{r} = \langle v,x \rangle$$locally uniformly implies that the derivative of $f$ at $0$ exists and $Df(0) = v$.I don't follow how...
View ArticleWhy is it that...
I mean, I know that it is true because by graphing each of the functions for each $N$ gives me that the difference is as small as I want. But why is that? How can I answer to this question? I think...
View ArticleConvergence of series using uniform continuity
Let $f:(0,+\infty)\rightarrow R$ be uniformly continuous. Prove that infinite series : $\sum\frac{1}{n}(f(n)-2f(n+1)+f(n+2))$ is convergent.I saw some behaviour of this sum, and that for some first...
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Real analysis sequence convergence problem
Sorry for not using LaTeX, i don't know how to use ithere's my problemi should mention that the sequence $\frac{m_n}{k_n}$ is not a constant sequence.$\text{If } \left\{ \frac{m_n}{k_n} \ne b \right\}...
View Articleif function is continuous then show that [closed]
Let $f : [0, 1] \to \Bbb R$ be a continuous function. Show that$$\lim_{n\to \infty} \prod_{k=1}^n\left(1+\frac 1nf\left(\frac kn\right)\right)=\exp\left({\int_0^1f(x)\,dx}\right)$$$$let\ y = \lim_{n\to...
View ArticleGeneral version of the fundamental lemma of calculus of variations
Let the measure $\mu$ on $\mathcal{B}(\mathbb{R}^n)$ be a general Borel measure.Let $f:\mathbb{R^n} \to \mathbb{R}$ be a locally integrable function and $\int f\varphi d\mu =0$ for any smooth function...
View ArticleA property of $C^1(\mathbb{R})$ functions
Let $f\in C^1(\mathbb{R})$ and let $x_0\in P$ where $P$ is a perfect set.I want proof that $\forall \varepsilon>0$$\exists\delta>0$ with $\lvert f(x)-f(y)-f'(y)(x-y) \rvert<\varepsilon\lvert...
View ArticleHow to evaluate the general limit of $\lim\limits_{n \to \infty...
I encountered this problemIf $f$ is a is continuously differentiable on $[0,1]$, find $$\lim_{n \to \infty }n\left(\sum_{k=1}^n \frac{1}{n}f\left(\frac k n\right)-\int_0^1f(x)dx\right)$$My attempt...
View ArticleIf $f$ is injective, what do we know about $f'$?
If $f$ is injective on an open interval $(a,b)$, then what properties we can deduce about the zeros of $f'$?We can not get that $f' \neq 0$. One example is $f(t) = t^3$ and $f'(0)=0$. I think at least...
View ArticleExistence of antiderivative of $f^2$
Let $f: \mathbb{R} \to \mathbb{R}$ have antiderivative. Does $f^{2}$ also have antiderivative? I know that the converse is false for example take $f(x)=1$ for rational $x$ and $f(x)=-1$ for irrational...
View ArticleEuler-Lagrange equation for the function of the integral
Can I derive an Euler-Lagrange equation for the following functional:$$F[u] = \phi(\int L(u,u^{'},x) dx)$$with constraint$$\int L(u,u^{'},x) dx = c \quad \forall c \in \mathbb{R} $$The function...
View ArticleFind real solutions to equation
I would like to find symbolic expressions for $x$ which satisfy the equation$$-(a \, x^2 - 2\, e\, d \, x)(a+b)\sqrt{a\, b\, (a^2\, x^2 - 2\, a\, d\, e\, x - c\, d)} + 2\, b$$$$\,(a\, x - e\,...
View ArticleDetermine $f:\mathbb R \to \mathbb R$ s.t. $f(x+y) = 2^xf(y)+2^yf(x)$
Let $f:\mathbb R \to \mathbb R$ be a continuous function such that f(1) = 1 and satisfies$$f(x+y) = 2^xf(y)+2^yf(x)\,\,\,\forall x \in R,\,\,\forall y \in R.$$Determine $$1.)\lim_{x \to 0}...
View ArticleFind the critical points of $x^2+y^2+z^2+2xyz$
Find the critical points of the function and specify the nature ofthese points. $$f(x,y,z)=x^2+y^2+z^2+2xyz$$I already found the critical points $A(0,0,0), B(1,1,-1), C(1,-1,1), D(-1,1,1),...
View ArticleIs this Dirichlet type function Riemann Integrable?
\begin{cases}\cos\left(\frac{\pi}{x}\right) &,\quad \text{if } x \text{ is rational}\\0 &,\quad \text{if } x \text{ is irrational}\end{cases}Over the interval $[0,1]$My approachSolve for upper...
View ArticleIs it possible to find a closed form for $ \Gamma(a-x)$ in terms of...
I wonder if there is a closed form for $x! (a-x)!$? or lets focus on $\Gamma(x) \Gamma(a-x)$ instead of the factorial.$$B(x,y)= \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$so $\Gamma(x) \Gamma(a-x)=...
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