Proving that there is no continuous function $f:\Bbb R\to\Bbb R$ satisfying $f(\Bbb Q)\subset\Bbb R-\Bbb Q$ and $f(\Bbb R-\Bbb Q) \subset\Bbb Q$.
234
No continuous function switches $\mathbb{Q}$ and the irrationals
113503
Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.
83338
Show that there's no continuous function that takes each of its values $f(x)$ exactly twice.
158221
Prob. 4 (a), Sec. 20 in Munkres' Topology, 2nd ed: Are these functions continuous in the product, uniform, and box topologies?
53738
Monotone functions and continuity
43189
Examples of continuous functions $f : \mathbb{R} \to \mathbb{R}$
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
131992
A problem on continuous functions
80872
Remark from baby Rudin about continuous function
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
36227
Show continuity or uniform continuity of $\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | )$
129716
Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
7876
If $f(x+y)=f(x)+f(y) ,\forall\;x,y\in\Bbb{R}$, then if $f$ is continuous at $0$, then it is continuous on $\Bbb{R}.$
24017
Let $f: \Bbb R \to [0,\infty)$ be a continuous function such that $g(x)=(f(x))^2$ is uniformly continuous. Which of the following is always true?
141223
$f(x^2) = 2f(x)$ and $f(x)$ continuous
145017
Example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points
27800
For continuous function $ f:\mathbb S^1 \to \mathbb R$ there exists uncountably many distinct points $x,y$ such that $f(x)=f(y)$