A description of all continuous functions on the quotient space. Is there any method which allows us to describe all continuous functions (maps to $\mathbb{R}$) on the quotient space?
For examle, how could I classify all continuous functions on $\mathbb{R}/[x\sim2x]$?
 A: You can classify them easily: these are the constant functions, whatever Hausdorff topological space $Z$ they land in.
If $f:\mathbb{R}/[x\sim 2x]\longrightarrow Z$ is continuous, then
$$
g(x):=f(\bar{x})
$$
is continuous on $\mathbb{R}$ by composition with the canonical surjection $x\longmapsto \bar{x}$ from $\mathbb{R}$ onto $\mathbb{R}/[x\sim 2x]$.
Then for every $x$
$$
g(x)=g\left(\frac{x}{2}\right)=\ldots=g\left(\frac{x}{2^n}\right)\qquad\forall n\geq 1.
$$
Letting $n$ tend to $+\infty$ and using continuity at $0$, we get
$$
g(x)=g(0)\quad\forall x\in\mathbb{R}.
$$
So $g$, hence $f$ is constant.
The converse is clear: every constant function on the quotient space is continuous.
A: Let $Y=X/\sim$ be a quotient space equipped with the quotient topology. $f:Y\to \mathbb{R}$ is continuous if and only if $f \circ \pi$ is continuous, where $\pi:X\to Y$ is the natural quotient map. The reason is $U\subseteq Y$ is open in the quotient topology if and only if $\pi^{-1}(U)$ is open in $X$.
A: Let's say that your quotient is describe by a relation $\rm R$. Then the Universal Property of the quotient topology tells you that there is a bijection $$ \text{ continuous functions on } \mathbb R/\mathrm{R} \longleftrightarrow \mathrm{R}-\text{invariant continuous functions on } \mathbb R.$$
Where $\rm R$-invariant means that $f(x) = f(y)$ when $x \sim_{\rm R} y$.
