I remembered one of my stupid constructions of my tender adolescence, being a pure product of the self-taught tradition. Oh, the beautiful time when one walked with one's feet on the sand and one's head in the stars while singing dramatic poems in honor of the ancient Greeks ... Here is an affine application: f (x) = a * x + b I chose two values of this application (x [1], x [2]) to determine a and b: solve([a*x[1]+b=f(x[1]),a*x[2]+b=f(x[2])], [a,b]); we are getting: a=(f(x[1])-f(x[2]))/(x[1]-x[2]), b=-(f(x[1])*x[2]-x[1]*f(x[2]))/(x[1]-x[2]) If we take two other points of this affine application, namely: (x, x [o]), we will have: solve([a*x+b=f(x), a*x[0]+b=f(x[0])], [a,b]); which gives: a=(f(x)-f(x[0]))/(x-x[0]), b=( x*f(x[0])-f(x)*x[0])/(x-x[0]) Noticing the result of a to which we can associate the derived function f '(x[0])=limit((f(x)-f(x[0]))/(x-x[0]), x, x[0]); At the result of b, I associated a new function that I named complementary function. ' f(x[0])=limit((x*f(x[0])-f(x)*x[0])/(x-x[0]), x, x[0]); I wrote the complementary function adding a "t" to differentiate with the derived function. ' f (x) = dt (f (x)) / dx; partial t (f (x, y) / partial (x); General formula: I came back to my affine application: a * x + b = f (x) by replacing a and b by its values: f(x)=x* (f(x)-f(x[0]))/(x-x[0])+ ( x*f(x[0])-f(x)*x[0])/(x-x[0]); I introduce the limits: limit(f(x) , x, x[0])= limit(x* (f(x)-f(x[0]))/(x-x[0]) , x, x[0])+ limit( ( x*f(x[0])-f(x)*x[0])/(x-x[0]) , x, x[0]); We come to the conclusion: f (x [0]) = ' f (x [0]) + x [0] * f ' (x [0]) Leading to the following formula that seems to me to connect: continuity, differentiability and complementarity. f (x) = ' f (x) + x f ' (x) = dtf (x) / dx + x * df (x) / dx A little silly the construction, it must be recognized, but I liked the complementary geometric operator. What is the point??