Is there a characterization of all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying:
My intuition is that $f$ should admit an integral representation: $$ f(t) = \int_0^t\, g(s)\,ds $$ for some positive continuous function to satisfy 1 and 2, but I don't know what would be needed for 3?
$\newcommand\R{\mathbb R}$Since $f$ is nondecreasing, concave, and continuous on $[0,\infty)$, it has a nonnegative nonincreasing right derivative $g=f'_+$ on $[0,\infty)$, which is of course right continuous. So, there is a limit $a_g:=\lim_{u\to\infty}g(u)\in[0,b_g]$, where $b_g:=g(0)<\infty$.
Let $\mu_g$ be the finite Lebesgue--Stieltjes measure on $(0,\infty)$ defined by the condition $\mu_g((u,\infty))=g(u)-a_g$ for $u\in(0,\infty)$; such a measure exists because $g$ is right continuous. Then for all $x\in[0,\infty)$ $$ \begin{aligned} f(x)&=\int_0^x du\,g(u) \\ &=\int_0^x du\,\Big(a_g+\int_{(u,\infty)}\mu_g(dv)\Big) \\ &=a_g x+\int_{(0,\infty)}\mu_g(dv)\int_0^{\min(x,v)} du \\ &=a_g x+\int_{(0,\infty)}\mu_g(dv) f_v(x), \end{aligned} $$ where $$f_v(x):=\min(x,v).$$
Vice versa, since for any $v\in(0,\infty)$ the function $f_v$ on $(0,\infty)$ is nondecreasing, concave, and continuous, and $f_v(0)=0$, we have the following: if $$f(x)=a x+\int_{(0,\infty)}\mu(dv) f_v(x) \tag{1}\label{1}$$ for some real $a\ge0$, some finite measure $\mu$ on $(0,\infty)$, and all real $x\ge0$, then the function $f$ on $(0,\infty)$ is nondecreasing, concave, and continuous, and $f(0)=0$.
We conclude that the functions $f$ on $[0,\infty)$ that are nondecreasing, concave, and continuous, and with $f(0)=0$ are precisely the functions of the form \eqref{1}.
Any increasing concave function $f:\mathbb R\to \mathbb R$ has a nonnegative derivative up to a countable set $N\subset \mathbb R$. Moreover $f$ is locally Lipschitz, thus locally absolutely continuous, so that $f(x)=\int_0^x f’(t)dt$, even as a Riemann integral, because $f’$ is a decreasing function. Conversely, every non-negative decreasing function $g:\mathbb R\to \mathbb R$ gives an increasing concave integral function $f(x)=\int_0^x g(t)dt$.
