Thomae function is continuous at $x\notin\mathbb Q$, discontinuous at $x\in\mathbb Q$
24817
Prove continuity on a function at every irrational point and discontinuity at every rational point.
157347
continuity, discontinuity derivative and relation to being derivative but its partials are not continuous
18222
Set of continuity points of a real function
143174
property of continuous function
81223
Real Analysis Question: continuity of $f(x) = \sum_{q_n<x} 1/n^2$
72044
prove $x \mapsto x^2$ is continuous
85623
Proof of continuity implied by differentiability
59519
What about the continuity of these functions in the uniform topology?
53738
Monotone functions and continuity
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
58664
Proving a function is continuous for a fixed variable
34940
Continuity of $f(x) = x^n$
40209
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
3772
About the continuity of the function $f(x) = \sum\limits_k2^{-k}\mathbf 1_{q_k \leq x}$
19916
$f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$
130586
$f$ is discontinuous $\iff$ kernel($f$) is dense in X
71487
Real Analysis: Continuity of a Composition Function