There cannot exist any non constant continuous function $f: \mathbb{R} \rightarrow \mathbb{Q}$
33330
Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$
113450
Construction of a function which is not the pointwise limit of a sequence of continuous functions
77943
If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$
73594
Existence of continuous function $f$ on $\Bbb R$ which vanishes exactly on $A\subset \Bbb R$
14994
Showing a function $f$ cannot be surjective
68165
Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point?
135199
Looking for a function $f$ that is $n$-differentiable, but $f^{(n)}$ is not continuous
90565
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
131992
A problem on continuous functions
132478
Prove that continuous functions mapping irrationals to rationals must be constant
143174
property of continuous function
82280
Problem about continuous functions and the intermediate value theorem
103241
A function that is continuous at the origin but not differentiable there
30036
Function in $H^1$, but not continuous
120947
continuous real-valued function
134720
Show that there is no increasing function $f: \aleph_1 \to \mathbb{R}$
157821
Let $S$ be a finite set. Show that any function $f: S \rightarrow \mathbb{R}^n$ is continuous on S. What if $S$ is countably infinite?