I think you can explain it even more clearly with smooth relative shifts, rather than sharp bump functions. I made a quick demo: https://www.desmos.com/calcula

I think you can explain it even more clearly with smooth relative shifts, rather than sharp bump functions. I made a quick demo: https://www.desmos.com/calculator/rfaqogkbmy https://www.desmos.com/calculator/rfaqogkbmy Drag the sliders for w and n to change how step-like the sigmoids are and how many are combined. The purple lines are the sigmoids, relative changes at each (regularly spaced) position, which are added together to make the blue approximation to the red function. You can change the function f(x) to see how it handles other possibilities, including piecewise/discontinuous ones like "{x<.5: 0, x>=5: x}".