Surely the following concept must have been thought of:
The extended-derivative of the step function f(x)={x<0 : 0 ; x>=0 : 1 } shall be defined as g(x)={x≠0:0; x==0:(0,1)}
Where (0,1) is a "Step Number" keeping track of the step.
Integral of g at 0 in interval ±ε is 1 for all ε>0.
(Higher order derivatives would lead to higher order "Step Numbers" (0,0,1) etc.)
Surely this must have been invented before and thus have a name? Or would such a system lead to irreconcilable contradictions?