Every continuous open mapping from $\mathbb{R}$ into $\mathbb{R}$ is monotonic
47
Let $f$ be a continuous and open map from $\mathbb R $ to $\mathbb R$.Prove that $f$ is monotonic.
12490
A map is continuous if and only if for every set, the image of closure is contained in the closure of image
128911
Continuous extension of a real function
74330
$\mathbb{R}^n$ not homeomorphic to open subset of $\mathbb{R}^m$
147788
Why is every map to an indiscrete space continuous?
147546
$f: \mathbf{R} \rightarrow \mathbf{R}$ monotone increasing $\Rightarrow$ $f$ is measurable
15095
continuous onto map from $(0,1)\to (0,1]$
36149
A uniformly continuous function maps bounded set to bounded sets
153495
Continuous function does not map closed set to closed set
121957
Is $p\mapsto \|f\|_p$ continuous?
103932
How can I prove that a polynomial with degree $n$ is continuous everywhere in $\mathbb{R}$ using definitions?
110957
No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof?
65472
From injective map to continuous map
133422
How to prove the continuity of an induced map; $g_* : C(X, Y) \rightarrow C(X, R)$, induced from the map $g \in C(Y, R)$, endowed with fine topology.
76234
Is this vector-valued map Hölder-continuous?
128069
Continuity in Slices + Mapping Compact Sets to Compact Sets = Continuous
52388
The equivalence of open-set definition of continuous mappings to the $\epsilon-\delta$ definition