Are bounded monotone functions piecewise continuous or at least their Laplace...
Suppose you have a function $h: \mathbb{R}_{+} \to \mathbb{R}_{+}$ that is monotone decreasing (thus bounded) and locally integrable. Can we conclude that it is piecewise continuous?Obviously, due to...
View Articlenonempty subset $E$ of $R$ closed and bounded iff every continuous...
I need to show that a nonempty subset $E$ of $R$ is closed and bounded iff every continuous real-valued function of $E$ takes a maximum value.I believe that "if $E$ is closed and bounded, then every...
View ArticleA question about a proof of "Optimal Control" by R. Vinter
I don't understand the proof of the Proposition 2.3.4. in the book "Optimal Control" by Richard Vinter.Proposition 2.3.4.: Consider a function $\phi\colon I\times \mathbb{R}^{n} \times\mathbb{R}^{m}\to...
View ArticleCan converge almost sure pass through nice function?
Suppose $f(x,y)$ is a continuous function, and $c$ is a given finite constant. Fixing $x$, $f(x,y)$ is strictly increasing with respect to $y$, and there exists a unique $y_x$ such that...
View ArticleProving $\lim_{(x,y)\to(\pi,\pi/2)} \frac{\cos y-\sin 2y}{\cos x\cos y}=1$...
I'm trying to prove the limit$$\lim_{(x,y)\to(\pi,\pi/2)} \frac{\cos y-\sin 2y}{\cos x\cos y}.$$Since$$\frac{\cos y-\sin 2y}{\cos x\cos y} = \frac{\cos y - 2\sin y\cos y}{\cos x\cos y} = \frac{\cos y(1...
View Articleoutside parallel curves converge uniformly to original curve with given...
Suppose we have a continous curve $\gamma:[a,b]\rightarrow\mathbb{R}^2$ where all one-sided derivatives exist. By $\gamma_\epsilon:[a,b]\rightarrow\mathbb{R}^2$ we mean the outside shifted parallel...
View ArticleMetric space with every infinite set having a limit point
I am trying to prove that a metric space in which every infinite subset has a limit point is compact. I am trying to prove it with Heine-Borel theorem for general metric spaces, since I have not...
View ArticlePartial derivative of $f(u,v)$
Let $f(u,v) = c$ where $u(x,y) , v(x,y)$ are functions and $c$ is constant. Can we conclude $\frac{\partial f}{\partial v} = \frac{\partial f}{\partial u} = 0$ ? It really sounds confusing to me but...
View ArticleIs This Approach Feasible?
Given a finite set of input-output pairs, is it possible to restore the function using Fourier transform-based interpolation, and then approximate the function with a polynomial that is easier to...
View ArticleCharacterizing the asymptotic properties of $f(k)>\frac{ak^2}{k-1}$
Context: Let $a>0$ be some given constant. Let $f:\{2,3,\text{...}\}\to\mathbb{R}_+$ be some increasing function. Consider the following inequality:$$\qquad f(k)> a\frac{k^2}{k-1}. \tag{$*$} $$I...
View ArticleSurjectivity of the gradient of a convex function
Question: Let $f: \mathbb{R}^{d} \to \mathbb{R}$ be a smooth strictly convex function with the growth condition $|f(x)|/|x| \to \infty \ (|x| \to \infty)$. Is the gradient $\nabla f$ surjective?My...
View ArticleProof of the Asymptotic Bound for the Definite Integral $\int_0^{\theta}...
I am investigating the asymptotic behavior of the integral $\int_0^{\theta} \sin^n x \, dx$ for any positive integer $n$ and $\theta \in [0, \pi]$. Current research, such as the discussion on...
View ArticleProperty of an ODE solution.
Let y be a solution of the following equation on $\mathbb{R}$: $y'(t)= \sin(ty(t))$I want to proof that: y is constant $\iff y(0)=0$I started by proofing: y is constant $\Rightarrow~ y(0)=0$.Let y be...
View ArticleIf $\sum a_n$ is divergent , $b_n \uparrow \infty $ then $\Sigma a_n b_n $ is...
If $\sum a_n$ is divergent, $b_n$ is unbounded and increasing then $\sum a_n b_n $ is divergent?I think it is true.Given $\sum a_n$ is divergent, so the sequence of partial sums $S_n=\sum_{k=1}^n a_k $...
View ArticleDomain of a function and simplification of fractions
I was working on the following exercise from a math book:Question: what is the domain of the following real function:$$ f(x)= \frac{x-1}{(x^2-1)(x^2-7x+10)} $$Step 1, I develop:$$...
View ArticleDoes there exist a continuous function that changes sign and monotonicity at...
I'm looking for a function with the following properties, and I'm wondering whether such a function exists. Let $I$ be an open interval, and let $c \in I $ be a point within that interval. I would like...
View ArticleMinimum of a continuous function on the interior of the unit ball
I consider the set $S = \{x\in\mathbb{R}^n : \lVert x\rVert < 1\}$ and $f : S\to\mathbb{R}$ a continuous function satisfying$$f(x)\geq\frac{1}{1-\lVert x\rVert}\quad (1)$$The problem I want to solve...
View ArticleWhy does the definition of limit imply these two facts in the result on the...
I have two potentially related questions concerning limits. I'm reading 2.48 in Measure, Integration & Real Analysis by Sheldon Axler, which is reproduced below.Why does the definition of limit...
View ArticleWhere the composition of limits is not always valid
Suppose we have a function $ f: X \to Y $ such that $ d_1 (f (x), y_0) <A_1 $ whenever $ d_0(x, x_0) <A_0 $ and so such a function $ g:Y \to W $ such that $ d_2 (g (y), w_0) <A_3 $ if $ d_1...
View Articlewhat $a_i$ ensures that $\sum^{\infty}_{i=0}a_i^2 < \infty$?
What requirements are needed on $a_i$, a non-negative sequence to ensure that $\sum^{\infty}_{i=0}a_i^2 < \infty$?My answer:If the series converges then it must be that $a_i^2$ approaches $0$, so...
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