Prove or Disprove: There exists $f: \mathbb{R} \rightarrow \mathbb{R}$ that is onto and one-to-one but not continuous.
118192
Is there a bijective function $f: \mathbb{R}\to\mathbb{R}$ that is discontinuous?
143354
Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
52798
Finding counterexamples: bijective continuous functions that are not homeomorphisms
73594
Existence of continuous function $f$ on $\Bbb R$ which vanishes exactly on $A\subset \Bbb R$
125043
Show that F(A−B) is not necessarily a subset of F(A)−F(B).
44144
If $f:[a,b]\to \mathbb R$ is continuous then so is $\max_{t\in [a,x]}f(t)$?
49592
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.
62817
Functions which are Continuous, but not Bicontinuous
98772
Continuity of a function in $\mathbb{R}$
40209
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
145077
A continuous bijection $f:\mathbb{R}\to \mathbb{R}$ is an homeomorphism?
47251
Non-compactness of $\mathbb{R}$ with the cocountable topology
117502
Proving that a certain continuous function is surjective.
118332
proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
136011
Must $f$ be continuous?