non linear staircase function

Question

Given that (not sure where this function comes from):

$S\left(x\right)=H\left(\frac{x}{W}+\frac{1}{2}-\frac{\arctan\left(\tan\left(\pi\left(\frac{x}{W}+\frac{1}{2}\right)\right)\right)}{\pi}\right)$

(Height=H and Width =W of step)

makes a reasonable staircase function but with equal steps.

Can this be adapted so the steps are quadratic (x,y)->(0,0) (1,1) (2,4) instead of (x,y)->(0,0) (1,1) (2,2)?

I wonder if it possible to adapt or create using only fundemental functions$^*$ - or whatever they are called I forget the name but functions like sine e log ect - as was done above?

*ie not using Heavyside, using modulas, floor ect.

Answer 1

$$S\left(x\right)=\text{sgn}(x) H\left(\frac{x}{W}+\frac{1}{2}-\frac{\arctan\left(\tan\left(\pi\left(\frac{x}{W}+\frac{1}{2}\right)\right)\right)}{\pi}\right)^2$$

$\text{sgn}(x)$ is the sign of $x$. This might be superfluous since the function is not specified for $x<0$ in the wording of the question.