$f: \Bbb R\to\Bbb R$ maps every open subsets to open subsets, then $f$ may not be continuous?
17607
Open maps which are not continuous
51156
Function is Continuous When Restricted to Closed Sets Which Cover the Space
139612
Follow-up regarding right-continuous $f:\mathbb{R} \to\mathbb{R}$ is Borel measurable
151427
f, g continuous for all rationals follow by continuous on all reals?
62817
Functions which are Continuous, but not Bicontinuous
44402
If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.
28111
Let $f:[1,10]\to \Bbb{Q}$ be a continuous function and $f(1)=10,$then $f(10)=?$
27800
For continuous function $ f:\mathbb S^1 \to \mathbb R$ there exists uncountably many distinct points $x,y$ such that $f(x)=f(y)$
135661
Weakly continuous operators are continuous
129087
Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$
144614
If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
98772
Continuity of a function in $\mathbb{R}$
77621
Continuity of function mapping connected set to connected set
33330
Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$
12490
A map is continuous if and only if for every set, the image of closure is contained in the closure of image
157603
Continuity $\iff$ Preimage is closed whenever set is closed
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$