Solving for $f$ when $f(x+y)=f(x)+f(y)+axy$ where $a$ is a real number
This question has been asked before here but I have followed a different approachWe have:$$f(0) = 0$$$$f(x) = f(\frac{x}{2}+\frac{x}{2}) = 2f(\frac{x}{2})+a\frac{x^2}{4}$$$$f(\frac{x}{2}) =...
View ArticleHeine borel theorem doesn't hold
Let $X$ be a metric space with underlying set $\mathbb{R}$ and metric $d(x,y)=\min\{|x-y|,1\}$. We have to find a subset of $X$ which is closed and bounded but not compact.My attempt:Since...
View ArticleTight bound proof
Suppose that for a classifier, the worst-case error rate is shown to be $\leq \frac{1}{4}$. I want to prove that $\frac{1}{4}$ is a tight upper bound.To do so, my approach is that it is sufficient to...
View ArticleProve $f$ is continuous at $x = 0$ in the topological way
I want to prove that $f: [0, +\infty) \to \mathbb{R}$ defined by $f(x)= \sqrt{x}$ is continuous at $x = 0$ in the topological way, that is (from Wikipedia): $F: X \to Y$ is continuous at $x = x_0 \in...
View Articlecollection of continuous functions having certain property.
Let $f_n : \mathbb{R} \to\mathbb{R}$ be continuous functions for n ≥ 1. Suppose foreach $x \in \mathbb{R}$, there exists $n(x) \in\mathbb{ N}$ such that $f_{n(x)}(x) = x + n(x).$Prove that there exists...
View ArticleInverse higher order derivatives of a function with respect to changes in...
I have an optimization problem operating on the implicit function $$F(x, y, z) = 0$$ and a vector $$(u, v, w)$$where the solution to the problem is the point $(x_0, y_0, z_0)$ that satisfies the...
View Articlefundamental theorem of calculus extended
Currently took calculus, and so I am familiar with the fundamental theorem of calculus... but I'm not sure how to do:$$\frac{d}{dx} \int_{0}^{\sqrt{\frac{x}{2}}} f(y) d\pi(y)$$It seems like an...
View ArticleSumming up Fourier coefficients $\hat{f}({\bf m})$ of $f\in L^1({\bf T}^n)$...
Say $f\in L^1({\bf T}^n)$ a function on the real n-torus ${\bf T}^n$ with $\sum_{{\bf m}\in {\bf Z}^n} \mid \hat{f}({\bf m})\mid<\infty$ where $\hat{f}({\bf m})=\int_{{\bf T}^n} f({\bf x}) e^{-2\pi...
View ArticleContinuation of a function
if I have a function $f(t)=e^{it}g(t)$, where $t=x\cdot \xi$ for $x,\xi \in R^n$. What are the conditions for defining $exp(it)= f(t)/g(t)$. The function $g$ is zero on a set with zero measure.The...
View ArticleContinuous function which satisfies the Luzin N property, but which does not...
My question is: can we find the function $g\in{\rm C}([a,b])$ which satisfies the Luzin N property on $[a,b]$, but which does not satisfy the Banach S property on [a,b]? Here $[a,b]$ is a compact...
View ArticleOn continuous functions and convergent sequence
The function $f : (0, 1] \to\mathbb R$ is a bounded and continuous on $(0, 1]$. Let $\{x_n\}$ be a sequence in $[0, 1]$. Prove or disprove the following.(a) If $\{x_n\}$ is convergent, then...
View ArticleStrictly positive linear functional: If there is one, there are infinitely many
Let $\mathbf{y}=\{Y_{n}\}_{n=1}^{\infty}$ be a sequence in aninfinite-dimensional Lebesgue space $L^{p}(\mathsf{\Omega},\mathcal{F},\mathrm{P})$, $p\geq1$, on some probability space...
View ArticleIs there an extension of the digit sum function to real numbers?
Is there a known way to extend the base $b$ digit sum function $s_b(\sum_{i=0}^{l-1}d_i b^i)=\sum_{i=0}^{l-1}d_i$ to a smooth function on the real numbers similarly to how one can extend the factorial...
View ArticleTrouble with proof that $L^p(\mathbb R ^n)$ is separable
I'm having trouble understanding the following proof from Linear Functional Analysis - An Application-Oriented Introduction (4.18, Proof for (4)):Let $\epsilon>0, 0<p<\infty$ and $f\in...
View ArticleThe continuous functions $f, g : \mathbb{R} \rightarrow \mathbb{R}$ satisfy...
(a) Using the definition of continuity, prove that $f(x) = g(x)$ for all $x \in \mathbb{R}$. (b) Use sequential criteria of continuity to redo the problem.I was able to do the part (b) of this problem...
View ArticleShowing that $\lim_{x\to...
In finding the rotational partition function for a $C_{\infty v}$ molecule, one comes across the function\begin{equation}Z(x)=\sum_{j=0}^\infty(2j+1)e^{-j(j+1)x}\end{equation}Surprisingly, this admits...
View ArticleProve that the solution operator is continuous
Let $\Omega\subset\mathbb R^n$ be open and bounded and let $f,g\in L^2(\Omega)$. Let $\phi\in H^1_0(\Omega)$. Let $\psi\in H^1_0(\Omega)$ be the weak solution of$$\int_\Omega \nabla \psi\cdot\nabla v...
View ArticleCan difference quotient sets be nowhere dense?
Let $f:\mathbb{R} \to \mathbb{R}$ be a function and consider the difference quotient set $$D_f = \left\{\frac{f(y) - f(x)}{y-x} : (x,y) \in \mathbb{R}^2, y > x\right\}$$Can $D_f$ be nowhere dense in...
View ArticleBounded seqeunce in $L^2(\Bbb R^d)$ has a weakly convergent subsequence
Let $\{f_n\}$ be a bounded sequence in $L^2=L^2(\Bbb R^d)$, i.e. $\sup_n ||f_n||_{L^2}<\infty$. I am trying to show that it is weakly convergent in the sense that there is $f\in L^2$ and a...
View ArticleDifference of suprema is less than or equal to supremum of difference [closed]
I can’t prove the following inequality:$$|\sup f-\sup g|\le\sup|f-g|$$in $[a,b]$, given $f,g:[a,b]\to\Bbb R$.
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