query: $F$ is a closed set in $\mathbb{R}$, Then $\frac{d(x+y, F)}{|y|}\rightarrow 0, |y|\rightarrow 0$ for each $x\in F$.
positive_passages.docid: 157577
positive_passages.text: Prove that $\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$ for a.e. $y \in F$.
negative_passages.docid: 152976
negative_passages.text: Real Analysis: open and closed sets
negative_passages.docid: 66293
negative_passages.text: $der(\{x\})$ is closed and $x\in der(der(A))$
negative_passages.docid: 80643
negative_passages.text: Show that f(x) defined on [0,infinity) R to is continuous at 0.
negative_passages.docid: 106944
negative_passages.text: Is the set defined by y=sin(1/x) open or closed?
negative_passages.docid: 1582
negative_passages.text: $f(x+1)-f(x)$ converges $\Rightarrow\frac{f(x)}x$ converges
negative_passages.docid: 85076
negative_passages.text: Is it true that $d(x,Y)=d(x,y_0)$ for some $y_0 \in Y$ when $Y$ is closed?
negative_passages.docid: 17646
negative_passages.text: If $\frac{f(x)}{x}$ is decreasing in $(0,\infty)$, prove that $f(a+b) \leq f(a) + f(b)$ for $a,b >0$
negative_passages.docid: 46287
negative_passages.text: Let $Y$ be an ordered set in the order topology with $f,g:X\rightarrow Y$ continuous, show that $\{x:f(x)\leq g(x)\}$ is closed in $X$
negative_passages.docid: 18530
negative_passages.text: Given 4 integers, $a, b, c, d > 0$, does $\frac{a}{b} < \frac{c}{d}$ imply $\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$?
negative_passages.docid: 40562
negative_passages.text: For $f :\mathbb{R} \rightarrow \mathbb{R}\displaystyle$ we have ${\lim_{n \to +\infty} f(x+n) =0}$ almost everywhere
negative_passages.docid: 27253
negative_passages.text: Path connectedness of the set $\{(x,y):(x+1)^{2}+y^{2}\leq 1\}\cup\{(x,y):y=x\sin(\frac{1}{x}),x>0\}$
negative_passages.docid: 82333
negative_passages.text: let $(X,d)$ be a metric space. How I can show that any finite subset of $X$ is closed.
negative_passages.docid: 123753
negative_passages.text: In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
negative_passages.docid: 97316
negative_passages.text: Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X
negative_passages.docid: 84510
negative_passages.text: If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
negative_passages.docid: 111485
negative_passages.text: Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$