Must a continuous function on $\mathbb R$ with only rational values be constant?
13765
Is a rational-valued continuous function $f\colon[0,1]\to\mathbb{R}$ constant?
151427
f, g continuous for all rationals follow by continuous on all reals?
102143
$f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $\int_{0}^{\infty}f(x)dx$ exists.
143798
Prove that, there exists no continuous function $f:\mathbb R\rightarrow\mathbb R$ with $f=\chi_{[0,1]}$ almost everywhere.
98666
Continuity defined for rational and irrational numbers
88586
show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant
234
No continuous function switches $\mathbb{Q}$ and the irrationals
84374
Rational function with absolute value $1$ on unit circle
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 $\mathbb R$
9406
Compactness of $(M, d)$ in terms of continuous functions $f:M\to \mathbb R$
45222
When $f(x) = g(y)$ for almost every $(x,y)$, must $f$ and $g$ be constant almost everywhere?
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
40576
Intermediate value property problem and continuous function
132113
Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
28120
Are intervally constant analytic functions always constant?
113503
Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.
11828
continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$