Equation for a smooth staircase function I am looking for a smooth staircase equation $f(h,w,x)$ that is a function of the step height $h$, step width $w$ in the range $x$. 
I cannot use the unit step or other similar functions since they are just one step. I have been experimenting with various sigmoid curves and while I can get a single smooth step I cannot get to realize the staircase shape. The closest staircase function I have found is given in this paper in equation (18) and depicted in Fig. 4 and it is a close example of what I want (i.e generate a staircase in the range $x$ for arbitrary step heights and widths) but it is not smooth at all.
Regarding smooth steps, a likely starting point I found is here but it gives a smooth function of just a single step. I have been unable to modify the equation to make it into a staircase. I would like to specify arbitrary step heights and widths and generate a smooth staircase in the range $x$ specified.  
Edit (Extra info):
The smooth function I mention above has the problem that the upper, horizontal line is not equal in length to the lower, horizontal line which is why I have been unable to adapt it into a staircase function 
Edit 2
Including some pictures 


Edit 2
Plot of $s$ with a steep slope showing a different width on the first horizontal line


Let $s : [0,1] \to [0,1]$ be a smooth function representing a single step. Assume that there exists some $\epsilon > 0$ such that $s(x) = 0$ for all $x < \epsilon$ and $s(x) = 1$ for all $x > 1 - \epsilon$. Setting
$$ f(x) = s(x - \lfloor x \rfloor) + \lfloor x \rfloor$$
then gives us a smooth staircase with steps of height and width $1$. By rescaling $f$, we can get steps of arbitrary width $w$ and height $h$:
$$f(h,w,x) = h f(x/w) = h(s(x/w - \lfloor x/w \rfloor) + \lfloor x/w \rfloor).$$
