A single-variable continuous function that is irrational if and only if its argument is rational
234
No continuous function switches $\mathbb{Q}$ and the irrationals
14251
$P[X=x]=0$ when $X$ is continuous variable
103241
A function that is continuous at the origin but not differentiable there
125812
Is sin(x) necessarily irrational where x is rational?
57180
Techniques to prove a function is uniformly continuous
118048
Can a function with just one point in its domain be continuous?
5419
Continuous functions with $f^2(x)=g^2(x)\neq 0$
53738
Monotone functions and continuity
21269
Bijective continuous function between $(0 ,\infty)$ and $\mathbb{R}$.
76396
Nowhere monotonic continuous function
58664
Proving a function is continuous for a fixed variable
12349
a construction of a continuous function from a normal space to [0,1]
87131
Discontinuous at rationals and differentiable at irrationals?
2321
Can there be two distinct, continuous functions that are equal at all rationals?
68165
Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point?
65895
Irrationality proofs not by contradiction
57233
Proof of continuity of Thomae Function at irrationals.