If $f:\omega_1 \to \mathbb{R}$ is continuous, then there is $a \in \omega_1$ such that if $x\geq a$, then $f(x) = f(a)$.
88586
show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant
145919
If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.
19916
$f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$
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 $\Bbb{R}.$
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
44574
Continuity of absolute value of a function
129073
If $f'$ is bounded, then $f$ is uniformly continuous
29154
$f$ is continuous and open implies $f$ injective
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
102457
for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.
30965
$f$ is continuous on $X$ iff $f$ is continuous on every compact subset
158423
How to prove that if $f$ is continuous a.e., then it is measurable.
2093
If $f$ is continuous then $G$ is homeomorphic to $X$.
112409
If $f$ is continuous and $f'(x)\ge 0$, outside of a countable set, then $f$ is increasing
9320
Extension of continuous function on a closed subset of $\mathbb{R}$
150554
proof that $f'$ is continuous at $a$