Does there exist a continuous function $f: \Bbb R\to \Bbb R$ that is rational at (Lebesgue) almost every irrational, and irrational at every rational?
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?
158221
Prob. 4 (a), Sec. 20 in Munkres' Topology, 2nd ed: Are these functions continuous in the product, uniform, and box topologies?
34836
Composition of Absolutely Continuous Functions
27800
For continuous function $ f:\mathbb S^1 \to \mathbb R$ there exists uncountably many distinct points $x,y$ such that $f(x)=f(y)$
53425
Prove that there exists a Borel measurable function $h: \mathbb R \to \mathbb R$ such that $g= h\circ f$.
24914
Does every continuous map from $\mathbb{Q}$ to $\mathbb{Q}$ extends continuously as a map from $\mathbb{R}$ to $\mathbb{R}$?
97619
Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere?
73594
Existence of continuous function $f$ on $\Bbb R$ which vanishes exactly on $A\subset \Bbb R$
99117
Existence of a sequence of positive continuous functions on $\mathbb R$ which is unbounded precisely on $\mathbb Q$
139612
Follow-up regarding right-continuous $f:\mathbb{R} \to\mathbb{R}$ is Borel measurable
80872
Remark from baby Rudin about continuous function
13765
Is a rational-valued continuous function $f\colon[0,1]\to\mathbb{R}$ constant?
76009
Can a continuous function from the reals to the reals assume each value an even number of times?
43189
Examples of continuous functions $f : \mathbb{R} \to \mathbb{R}$
47992
Is there a function $f\colon\mathbb{R}\to\mathbb{R}$ such that every non-empty open interval is mapped onto $\mathbb{R}$?
33330
Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$
93461
Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty.$Prove that $f$ is uniformly continuous.