query: If $|f(x)-f(y)|\geq \frac12|x-y|$, must $f$ be bijective?
positive_passages.docid: 22061
positive_passages.text: $|f(x)-f(y)|\geq k|x-y|$.Then $f$ is bijective and its inverse is continuous.
negative_passages.docid: 6023
negative_passages.text: Surjectivity of Composition of Surjective Functions
negative_passages.docid: 157268
negative_passages.text: Proving a bijection with absolute value
negative_passages.docid: 104757
negative_passages.text: If $f \circ g$ is surjective, then f is surjective
negative_passages.docid: 17310
negative_passages.text: Can $f(x)=|x|^3$ be differentiated
negative_passages.docid: 64866
negative_passages.text: Proving a Composition of Functions is Bijective
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: 142845
negative_passages.text: is $y = \sqrt{x^2 + 1}− x$ a injective (one-to-one) function?
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: 57026
negative_passages.text: $f(0)=f'(0)=f'(1)=0$ and $f(1)=1$ implies $\max|f''|\geq 4$
negative_passages.docid: 108704
negative_passages.text: Help with proof of $|x+y| \geq |x|-|y|$
negative_passages.docid: 22210
negative_passages.text: Points of discontinuity of a bijective function $f:\mathbb{R} \to [0,\infty)$
negative_passages.docid: 20614
negative_passages.text: Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex
negative_passages.docid: 21391
negative_passages.text: Proving Functions are Surjective
negative_passages.docid: 72856
negative_passages.text: Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
negative_passages.docid: 45710
negative_passages.text: Finishing a proof: $f$ is injective if and only if it has a left inverse
negative_passages.docid: 2414
negative_passages.text: If $f \in \operatorname{Hol}(D)$, $f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$, prove that $|f(0)| \leq \frac{1}{4}$