Equation for a smooth staircase function

Here is an example based on Math536's answer: Wolfram link

$$f(h,w,a,x) = h \left[\frac{\tanh \left( \frac{ax}{w}-a\left\lfloor \frac{x}{w} \right\rfloor-\frac{a}{2}\right)}{2\tanh\left(\frac{a}{2}\right) } + \frac{1}{2} + \left\lfloor \frac{x}{w} \right\rfloor\right]$$

Where h is the step height, w is the period, and a is the smoothness

We can start with a simple soft staircase function:

$$ f(x) = x - sin \space x $$

and then feed it into itself:

$$ y(x) = f(f(x)) $$

then again:

$$ y(x) = f(f(f(x))) $$

and again:

$$ y(x) = f^4(x) $$

As you can see, each iteration makes the "flat" part of the step longer, and the rise steeper.

The period and the height of each step is $ 2 \pi $, so multiply $x$ by $2 \pi / w$ and $y$ by $h / 2 \pi$ to reach your desired scale.

In reality, the curve is only truly flat (zero derivative) at the centre of each step — at every $ 2 \pi k $ — and only close to flat on either side of that point.

Configurability is limited: The softness of the step can only be specified in integer amounts (the number of times we reapply $f$ to itself), and it requires many/infinite applications to make the step really sharp.

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 $h$: $$f(h,w,x) = h f(x/w) = h(s(x/w - \lfloor x/w \rfloor) + \lfloor x/w \rfloor).$$