query: $f: [a, b] \mapsto [a,b]$, $f$ is monotonically increasing $\Rightarrow \exists~ x \in (a, b)$ such that $x = f(x)$
positive_passages.docid: 149390
positive_passages.text: Fixed point of a Function (monotonically increasing)
negative_passages.docid: 72694
negative_passages.text: Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
negative_passages.docid: 47135
negative_passages.text: Prove if a = b, then f(a) = f(b) for any function f (with natural deduction)
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: 29582
negative_passages.text: Continuous mapping $f: [0,1]\rightarrow (0,1)$ CSIR December $2013$
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: 46778
negative_passages.text: Prove that a recursive sequence is monotonically increasing
negative_passages.docid: 34077
negative_passages.text: Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f\big(x-f(y)\big)=f\big(f(y)\big)+x\cdot f(y)+f(x)-1$ for all $x,y\in\mathbb{R}$.
negative_passages.docid: 13852
negative_passages.text: Assume $f$ is a continuous one-to-one function over an interval. Prove that $f$ is strictly monotone
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: 37308
negative_passages.text: Let $f:K\to K$ with $\|f(x)-f(y)\|\geq ||x-y||$ for all $x,y$. Show that equality holds and that $f$ is surjective.
negative_passages.docid: 61640
negative_passages.text: Prove c exists for f:[0,1] -> R
negative_passages.docid: 109498
negative_passages.text: Linear maps $f: M_n \longrightarrow \mathbb{R}$ such that $f(AB) = f(BA)$
negative_passages.docid: 53551
negative_passages.text: $f(A\cap B)=f(A)\cap f(B)$ $\iff$ $f$ is injective.
negative_passages.docid: 152841
negative_passages.text: I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.
negative_passages.docid: 52323
negative_passages.text: Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$
negative_passages.docid: 72006
negative_passages.text: Suppose $f(x)$ is linear i.e. $f(x + y) = f(x) +f(y) $ and monotone on $[-\infty, +\infty]$, then $f(x) = ax$, a real.