First connect them with straight lines:
Now imagine you are considering the derivative:
The next step is to do what I'm calling triangularizing the above step function. This means to find the midpoints of AB and CD, E and F. Then find the point G that preserves the total area under the original step function.
This makes a new curve when you remove B and C,
Now this is a new approximation for the derivative, so we integrate and set the constant of integration so A goes to [1,1] where it was originally:
To get smoother and smoother approximation, iterate the derivative and triangularization step and integrate the corresponding number of times.