$\DeclareMathOperator\CM{CM}$
I recently came across [Okhezin - Study of families of monotone continuous functions on Tychonoff spaces](https://link.springer.com/content/pdf/10.1007/s10958-007-0259-2.pdf) describing monotone functions on general topological spaces and I wonder the following more basic question. Let $\CM^+(\mathbb{R})$ be the set of continuous, surjective, and strictly monotone functions (actually these must be homeomorphisms) from $\mathbb{R}$ to $\mathbb{R}$.


Has anyone come across any literature describing a basis of the infinite-dimensional space $\CM^+(\mathbb{R})$?



Basis for space of continuous, surjective monotone functions on $\mathbb{R}$
$\newcommand\R{\mathbb R}$$\newcommand\LBV{\mathrm{LBV}}$As was noted in comments, the set of all monotone functions (or your version of it,
$\mathrm{CM}^+(\R)$) is not a linear space.


An appropriate linear space here is $\LBV$, the space of all (say) right-continuous functions from $\R$ to $\R$ of locally bounded variation — because any function of locally bounded variation is the difference between two nondecreasing functions.


Take now any $f\in\LBV$. Let $\mu\_f$ be the corresponding signed Lebesgue–Stieltjes measure, defined by the formula $\mu\_f((a,b])\mathrel{:=}f(b)-f(a)$ for all real $a$ and $b$ such that $a<b$. It is easy to check that for all real $x$ we have
$$f(x)=f(0)+\int\_{(0,\infty)}\mu\_f(du)\,1(x\ge u)-\int\_{(-\infty,0]}\mu\_f(dv)\,1(x<v);$$
that is,
$$f=f(0)\,1+\int\_{(0,\infty)}\mu\_f(du)\,1\_{[u,\infty)}-\int\_{(-\infty,0]}\mu\_f(dv)\,1\_{(-\infty,v)}.$$


Thus, any $f\in\LBV$ is the limit of linear combinations of the following, say basic, functions: (i) the constant $1$, (ii) the functions of the form $1\_{[u,\infty)}$ for real $u>0$, and (iii) the functions of the form $1\_{(-\infty,v)}$ for real $v\le0$. In this sense, one might want to say that these basic functions form a basis of $\LBV$.


$\big($Because $1\_{(-\infty,v)}=1-1\_{[v,\infty)}$ for all real $v$, we have another, equivalent set of the basic functions of $\LBV$, consisting of the following functions: (i) the constant $1$ and (ii) the functions of the form $1\_{[u,\infty)}$ for all real $u$.$\big)$


