Prove that there isn't such a function, which would be continuous at each rational point and discontinuous in each irrational point.
18222
Set of continuity points of a real function
18511
Can we construct a function that has uncountable many jump discontinuities?
98666
Continuity defined for rational and irrational numbers
36983
Does there exist a continuous function such that f(x) is rational for every irrational x, and f(x) is irrational for every rational x?
38886
Discontinuous function on Q
124444
If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?
65895
Irrationality proofs not by contradiction
27800
For continuous function $ f:\mathbb S^1 \to \mathbb R$ there exists uncountably many distinct points $x,y$ such that $f(x)=f(y)$
113450
Construction of a function which is not the pointwise limit of a sequence of continuous functions
113503
Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.
3518
Proving $f(x)=1/x$ on $(0,1 )$ is not uniformly continuous
135199
Looking for a function $f$ that is $n$-differentiable, but $f^{(n)}$ is not continuous
57233
Proof of continuity of Thomae Function at irrationals.
25010
Argue that there exists a continuous function $g: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies $g(x) = \frac{1-e^{-2x^2}}{x^2}$ for $x \neq 0$
137883
Cardinality of the set of all (real) discontinuous functions
88259
prove that $\frac{1}{x}$ is not uniformly continuous on $(0,1)$
79107
Give an example of a function $f: \mathbb{R} \to \mathbb{R}$ which is continuous only at $0$.