query: There exists no continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ with $f=\chi_{[0,1]}$ almost everywhere
positive_passages.docid: 143798
positive_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: 112501
negative_passages.text: A funtion and its fourier transformation cannot both be compactly supported unless f=0
negative_passages.docid: 73594
negative_passages.text: Existence of continuous function $f$ on $\Bbb R$ which vanishes exactly on $A\subset \Bbb R$
negative_passages.docid: 158309
negative_passages.text: Find a function that is continuous only on rationals
negative_passages.docid: 95321
negative_passages.text: Subspaces of all real-valued continuous functions on $\mathbb{R}^1$
negative_passages.docid: 3922
negative_passages.text: How to prove that there exist no functions $f,g:\Bbb{R}\to\Bbb{R}$ such that $f(g(x))=x^{2018}$ and $g(f(x))=x^{2019}$?
negative_passages.docid: 57014
negative_passages.text: Existence of a power series converging non-uniformly to a continuous function
negative_passages.docid: 154572
negative_passages.text: Continuity of a function at an isolated point
negative_passages.docid: 103241
negative_passages.text: A function that is continuous at the origin but not differentiable there
negative_passages.docid: 58816
negative_passages.text: Existence of a continuous function which does not achieve a maximum.
negative_passages.docid: 112807
negative_passages.text: Show that no application $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, of $C^k$ class, $k \geq 1$ can be injective
negative_passages.docid: 25010
negative_passages.text: Argue that there exists a continuous function $g: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies $g(x) = \frac{1-e^{-2x^2}}{x^2}$ for $x \neq 0$
negative_passages.docid: 124535
negative_passages.text: continuous function from $[\pi,2\pi]\to \mathbb{R}^2$
negative_passages.docid: 97619
negative_passages.text: Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere?
negative_passages.docid: 148512
negative_passages.text: No analytic function $f$ has modulus $|f(z)|=1/\cosh(\Re z)$
negative_passages.docid: 83338
negative_passages.text: Show that there's no continuous function that takes each of its values $f(x)$ exactly twice.
negative_passages.docid: 69869
negative_passages.text: if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$