A function is discontinuous at all rational points and continuous at all irrational points
24817
Prove continuity on a function at every irrational point and discontinuity at every rational point.
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
46629
Prove if $f(x) = g(x)$ for each rational number x and $f$ and $g$ are continuous, then $f = g$
34836
Composition of Absolutely Continuous Functions
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
79107
Give an example of a function $f: \mathbb{R} \to \mathbb{R}$ which is continuous only at $0$.
87239
Prove functions defined by sup and inf are continuous
43001
Proof for Dirichlet Function and discontinuous
87131
Discontinuous at rationals and differentiable at irrationals?
152563
Set of discontinuity of monotone function is countable
71487
Real Analysis: Continuity of a Composition Function
32205
Topology on the real numbers such that all polynomials are continuous but cosine is not continuous
150833
Discontinuities of the derivative of a differentiable function on closed interval
89814
Limit of a Piecewise Function defined on Rationals and Irrationals
113503
Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.
158567
A uniform continuous function which is not Hölder continuous
2321
Can there be two distinct, continuous functions that are equal at all rationals?