Let $f,g$ be continuous on $\mathbb{R}→\mathbb{R}$ and $f(r)=g(r)$ if $r∈\mathbb{Q}$, is it true that $f(x)=g(x)$ for all $x∈\mathbb{R}$
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
126461
Show that $g:I\to\mathbb{R}$ is continuous and $f:(x-x_0)g(x)$ is differentiable
33330
Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$
40209
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
13765
Is a rational-valued continuous function $f\colon[0,1]\to\mathbb{R}$ constant?
101773
Let $g : A \to B$ and $f : B \to C$ be functions. Let $f \circ g$ be onto. Are both $f$ and $g$ necessarily onto?
45222
When $f(x) = g(y)$ for almost every $(x,y)$, must $f$ and $g$ be constant almost everywhere?
141223
$f(x^2) = 2f(x)$ and $f(x)$ continuous
136011
Must $f$ be continuous?
74950
Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
138105
Let $f$ and $g$ be continuous at $ x=a$, with $f(a)<g(a)$. Prove that there exists a neighborhood $I$ of a such that $f(x)<g(x)$ for all $x \in I$.
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
145077
A continuous bijection $f:\mathbb{R}\to \mathbb{R}$ is an homeomorphism?
2093
If $f$ is continuous then $G$ is homeomorphic to $X$.
83204
Is every $G_\delta$ set the set of continuity points of some function $f$?
144589
Proving that if $f(x) \le g(x)$ then $f(x_0) \le g(x_0)$
22061
$|f(x)-f(y)|\geq k|x-y|$.Then $f$ is bijective and its inverse is continuous.