query: No continuous injective map $f: \mathbb{S}^1 \to \mathbb{R}$
positive_passages.docid: 141835
positive_passages.text: Rigorous Proof: Circle cannot be embedded into the the real line!
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?
negative_passages.docid: 142174
negative_passages.text: Nonexistence of an injective $C^1$ map between $\mathbb R^2$ and $\mathbb R$
negative_passages.docid: 33129
negative_passages.text: A linear map $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$
negative_passages.docid: 15095
negative_passages.text: continuous onto map from $(0,1)\to (0,1]$
negative_passages.docid: 138481
negative_passages.text: $f(x)=x/1$ is not injective?
negative_passages.docid: 149095
negative_passages.text: $f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous
negative_passages.docid: 47251
negative_passages.text: Non-compactness of $\mathbb{R}$ with the cocountable topology
negative_passages.docid: 1117
negative_passages.text: Is there a bijection between $(0,1)$ and $\mathbb{R}$ that preserves rationality?
negative_passages.docid: 24828
negative_passages.text: surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism
negative_passages.docid: 72115
negative_passages.text: Why isn't there a continuously differentiable surjection from $I \to I \times I$?
negative_passages.docid: 151354
negative_passages.text: Why are there no non-trivial regular maps $\mathbb{P}^n \to \mathbb{P}^m$ when $n > m$?
negative_passages.docid: 143798
negative_passages.text: Prove that, there exists no continuous function $f:\mathbb R\rightarrow\mathbb R$ with $f=\chi_{[0,1]}$ almost everywhere.
negative_passages.docid: 4538
negative_passages.text: Is the mapping $ d : X\times X \mapsto \mathbb {R} $ continuous?
negative_passages.docid: 100736
negative_passages.text: Why every map $f : S^n \to T^n (n>1)$ has topological degree zero?
negative_passages.docid: 52798
negative_passages.text: Finding counterexamples: bijective continuous functions that are not homeomorphisms
negative_passages.docid: 27265
negative_passages.text: Continuously differentiable map from $\mathbb{R}^{m+n}$ to $\mathbb{R}^n$