Continous function iff $x \notin \mathbb{Q}$ or $x = 0$
102075
A function continuous on all irrational points
76396
Nowhere monotonic continuous function
115004
If $x\notin\mathbb Q$, then $\left|x-\frac{p}{q}\right|<\frac{1}{q^2}$ for infinitely many $\frac{p}{q}$?
69876
For every $n \in \mathbb{N}$, find a function $f$ which is differentiable $n$ times at $0$, but not $n+1$ times.
129087
Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$
134720
Show that there is no increasing function $f: \aleph_1 \to \mathbb{R}$
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$
40576
Intermediate value property problem and continuous function
92089
If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
132478
Prove that continuous functions mapping irrationals to rationals must be constant
17223
Continuity of an inverse function.
73594
Existence of continuous function $f$ on $\Bbb R$ which vanishes exactly on $A\subset \Bbb R$
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?
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 $x \neq 0$
8836
If $f$ is continuous on $[a , b]$ and $\int_a^b f(x) p(x)dx = 0$ then $f = 0$
133231
Discontinuity of Thomae Function on $\mathbb{Q}$
53738
Monotone functions and continuity