Is it possible to approximate every monotonic increasing function $f:[0,1]\rightarrow \mathbb{R}$ uniformly by a sequence $(f_n)_{n \in \mathbb{N}}$ of step functions on $[0,1]$, s.t. for all $n\in \mathbb{N}$ $f_{n+1}-f_n$ is monotonous increasing? Step functions denounces linear combinations of indicator functions of all intervals $[0,y],[0,y[, y \in \mathbb{R}$
It seems easy, but I'm starting to think, it's not possible at all and it seems to me, that's the case, because like for the identity I need $\mathcal{O}(\lceil\frac{1}{\varepsilon}\rceil)$ points to make for an error $<\varepsilon$ and the infinite sum of all the countably many corrections made before any given point, has to converge.
If $f$ is continuous and non constant, this is impossible.
Indeed, if $f_{n+1}-f_n$ is increasing for all $n \in \mathbb{N}$, then for all $p>q$, $f_p-f_q$ is increasing. Letting $p$ tend to $+\infty$, you get that $f-f_q$ must be increasing for all $q$. In particular, $f_q$ must be decreasing (otherwise you would get a contradiction at an increasing step of $f_q$, by continuity of $f$). But if $f$ is increasing and non constant, it cannot be the limit of decreasing functions.