query: Prove a real $f$ is injective if for all open sets $U$ in $\mathbb{R}$, $f(U)$ is open.
positive_passages.docid: 29154
positive_passages.text: $f$ is continuous and open implies $f$ injective
negative_passages.docid: 35378
negative_passages.text: Prove that $f$ analytic, $f(x) \in \mathbb{R}$ for all $x \in \mathbb{R}$ implies $f(\overline{z})=\overline{f(z)}$
negative_passages.docid: 25997
negative_passages.text: Conditions Equivalent to Injectivity
negative_passages.docid: 129081
negative_passages.text: Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic?
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: 93102
negative_passages.text: prove that $f:X\rightarrow Y$ is surjective if and only if $f(f^{-1}(C))=C$
negative_passages.docid: 99236
negative_passages.text: Interior of closure of an open set
negative_passages.docid: 102168
negative_passages.text: Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.
negative_passages.docid: 2575
negative_passages.text: Showing that $\mathbb{R}$ is connected
negative_passages.docid: 117502
negative_passages.text: Proving that a certain continuous function is surjective.
negative_passages.docid: 49882
negative_passages.text: Proving $f(x,y,z)=\big(\frac{x}{a+x+y+z}, \frac{y}{a+x+y+z}, \frac{z}{a+x+y+z} \big)$ is injective
negative_passages.docid: 72251
negative_passages.text: $g\circ f$ is Surjective and $g$ is Injective then Prove that $f$ is surjective
negative_passages.docid: 22232
negative_passages.text: Proving that cardinality of the reals = cardinality of $[0,1]$
negative_passages.docid: 20303
negative_passages.text: A function $f\colon E\to F$ between finite sets is injective iff it is surjective iff it is bijective
negative_passages.docid: 115636
negative_passages.text: Show that $f$ is either injective or a constant function.
negative_passages.docid: 14277
negative_passages.text: Proving $f(x)=x\cdot |x|$ is a bijection
negative_passages.docid: 126463
negative_passages.text: Can open sets and closed sets of $\mathbb{R}^n$ be homeomorphic?