query: If $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, 1-1, and on-to, then f maps Borel Sets to Borel Sets
positive_passages.docid: 23728
positive_passages.text: Does a continuous and 1-1 function map Borel sets to Borel sets?
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: 155618
negative_passages.text: Slightly changing the formal definition of continuity of $f: \mathbb{R} \to \mathbb{R}$?
negative_passages.docid: 97104
negative_passages.text: Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
negative_passages.docid: 122160
negative_passages.text: $f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
negative_passages.docid: 138815
negative_passages.text: For which conditions on countable sets does continuity implies uniform continuity
negative_passages.docid: 123432
negative_passages.text: If $f: \mathbb{C} \to \mathbb{C}$ is continuous and analytic off $[-1,1]$ then is entire.
negative_passages.docid: 9406
negative_passages.text: Compactness of $(M, d)$ in terms of continuous functions $f:M\to \mathbb R$
negative_passages.docid: 99117
negative_passages.text: Existence of a sequence of positive continuous functions on $\mathbb R$ which is unbounded precisely on $\mathbb Q$
negative_passages.docid: 132013
negative_passages.text: How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
negative_passages.docid: 49220
negative_passages.text: Open Sets of $\mathbb{R}^1$ and axiom of choice
negative_passages.docid: 55605
negative_passages.text: If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
negative_passages.docid: 101319
negative_passages.text: Properties of a continuous map $f : \mathbb{R}^2\rightarrow \mathbb{R}$ with only finitely many zeroes
negative_passages.docid: 125682
negative_passages.text: Proof verification: the product of two continuous functions is continuous
negative_passages.docid: 80643
negative_passages.text: Show that f(x) defined on [0,infinity) R to is continuous at 0.
negative_passages.docid: 130975
negative_passages.text: Given a pair of continuous functions from a topological space to an ordered set, how to prove that this set is closed?
negative_passages.docid: 7036
negative_passages.text: Is there a bijective continuous function $f: \Bbb R \to \Bbb R$ whose inverse $f^{-1}$ is not continuous?