query: Let $f: X \to X$ be such that $d(f(x), f(y)) = d(x, y)$ for all $x, y \in X$. To show that $f$ is onto.
positive_passages.docid: 152057
positive_passages.text: Let $(X,d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x),f(y)) = d(x,y)$ for all $x,y \in X$. Show that $f $ is onto (surjective).
negative_passages.docid: 14277
negative_passages.text: Proving $f(x)=x\cdot |x|$ is a bijection
negative_passages.docid: 71857
negative_passages.text: Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
negative_passages.docid: 121993
negative_passages.text: Prove that $f$ is one-to-one iff $f \circ h = f \circ k$ implies $h = k$.
negative_passages.docid: 6825
negative_passages.text: If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.
negative_passages.docid: 89514
negative_passages.text: Show that if $g \circ f$ is injective, then so is $f$.
negative_passages.docid: 130537
negative_passages.text: Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$
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: 45370
negative_passages.text: Is it true that $|d(x, z) - d(y, z)| \leq d(x, y)$ and how do I prove it?
negative_passages.docid: 122329
negative_passages.text: Determine all functions $f(x)$ such that $f(f(x+y))=f(x)+f(y)$
negative_passages.docid: 37966
negative_passages.text: If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
negative_passages.docid: 136842
negative_passages.text: $|x-f(x)|<5777$, Show that $f$ is surjective
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: 77074
negative_passages.text: If $ f(f(f(x)))=x$, does$ f(x)=x$ necessarily follow?
negative_passages.docid: 134703
negative_passages.text: Let $X$ be a connected space, $f:X\to X$ continuous involution, $g:X\to\mathbb{R}$ continuous. Prove that $\exists$ $x\in X$ such that $g(x)=g(f(x))$.
negative_passages.docid: 98151
negative_passages.text: Suppose that A is a countable set. Show that the set B is also countable if there is an onto function f from A to B.
negative_passages.docid: 1821
negative_passages.text: Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality