query: $f:\bf S^1 \to \bf R$, there exist uncountably many pairs of distinct points $x$ and $y$ in $\bf S^1$ such that $f(x)=f(y)$? (NBHM-2010)
positive_passages.docid: 27800
positive_passages.text: For continuous function $ f:\mathbb S^1 \to \mathbb R$ there exists uncountably many distinct points $x,y$ such that $f(x)=f(y)$
negative_passages.docid: 103019
negative_passages.text: Find all functoins $f: \mathbb R \rightarrow \mathbb R$ such that $\forall x,y \in \mathbb R f(xy+f(x))=xf(y)+f(x)$
negative_passages.docid: 152784
negative_passages.text: $f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.
negative_passages.docid: 45219
negative_passages.text: Proving that $C$ is a subset of $f^{-1}[f(C)]$
negative_passages.docid: 121395
negative_passages.text: Let $X$ and $Y$ be non-empty sets and let $f:X \rightarrow Y$. Prove that $ f^{ −1} :P(Y ) → P(X) $ is injective if, and only if, $f$ is surjective.
negative_passages.docid: 155575
negative_passages.text: Proving a polynomial $f(x)$ composite for infinitely many $x$
negative_passages.docid: 50012
negative_passages.text: $f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
negative_passages.docid: 72642
negative_passages.text: Continuous bijection from $(a,b) \to S^1$?
negative_passages.docid: 139168
negative_passages.text: Prove that there is an unique $z$ s.t. $f(z) = z$ where $z$ is a complex number
negative_passages.docid: 39321
negative_passages.text: If $f:\mathbb R \to \mathbb R$ satisfies $f'(x) <L$ for some $L<1$, then $f$ has a unique fixed point
negative_passages.docid: 49592
negative_passages.text: Show that, if $f:A\to B$ is a function, with $A$ and $B$ being finite sets, and $|A|=|B|$, then $f$ is one to one iff $f$ is onto.
negative_passages.docid: 22391
negative_passages.text: Is the set of all pairs of real numbers uncountable?
negative_passages.docid: 95976
negative_passages.text: Given a PRF $F$ , is $G(s) = F_s(1)\|F_s(2)\|\cdots \|F_s(n+1)$ a PRG?
negative_passages.docid: 92376
negative_passages.text: Let $f:A \rightarrow B$ and C ⊂A. Define f[C]={b∈B: b=f(a) for some a∈C}. Prove or disprove each of the statement
negative_passages.docid: 73850
negative_passages.text: Cardinality of set of functions
negative_passages.docid: 21431
negative_passages.text: The set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable?
negative_passages.docid: 88278
negative_passages.text: Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective