I'd need any (real-valued) function (whatever meets the following description at least approximately) continuous and thrice differentiable everywhere (or twice if 3 not possible), with the following features.
Seen "by far, it would be resembling the floor function $\lfloor x \rfloor $, exactly equal to $j$ at each positive integer $j$, but then slightly increasing on the "horizontal" part and finally more suddenly raise to the next integer, so approximately doing as follows. Let $\epsilon$ and $\alpha$ two arbitrarily small positive numbers given by the user. The function should be:
$j$ at any integer $j$ (I need only $j$ > 2, no need to consider negative integers)
increasing from $j$ to $j +\alpha$ near the the "horizontal part" of the floor, say approximately over $(j, j + 1 - \epsilon$].
suddenly increasing (or non-decreasing), from $j + \alpha$ to $(j + 1)$, "near the step" of a floor function [say approximately over $[j + 1 - \epsilon, j+1$).
These pieces should be "joined" in such a way that the function is continuous and differentiable. (Any function with the above characteristics will do: of course, the simpler the better: I need it for an iterative program)
PS. added later
If useful, I found this paper: http://math.arizona.edu/~shankar/projects/TermPaper_Yilu.pdf
which has something which perhaps may come close (see pag. 10/11). But it is missing the increasing "horizontal" part.