If Archimedes would have known functions

Could calculus on graphs have emerged by the time of Archimedes, if function, graph theory and matrix concepts were available 2300 years ago? 1. Single variable calculus Calculus on integers deals with functions f (x) like f (x) = x 2. The difference Df (x) = f (x) = f (x + 1) − f (x) = 2x + 1 as well as the sum Sf (x) = x−1 k=0 f (k) with the understanding Sf (0) = 0, Sf (−1) = f (−1) are functions again. We call Df the derivative and Sf the integral. The identities DSf (x) = f (x) and SDf (x) = f (x)− f (0) are the fundamental theorem of calculus. Linking sums and differences allows to compute sums (which is difficult in general) by studying differences (which is easy in general). Studying derivatives of basic functions like x n , exp(a·x) will allow to sum such functions. As operators, Xf (x) = xs * f (x) and Df (x) = [s, f ] where sf (x) = f (x + 1) and s * f (x) = f (x − 1) are translations. We have 1x = x, Xx = x(x − 1), X 2 x = x(x − 1)(x − 2). The derivative operator Df (x) = (f (x + 1) − f (x))s satisfies the Leibniz product rule D(f g) = (Df)g + (Dg)f = Df g + f + Dg. Since momentum P = iD satisfies the anti-commutation relation [X, P ] = i we have quantum calculus. The polynomials [x] n = X n x satisfy D[x] n = n[x] n−1 and X k X m = X k+m. The exponential exp(a · x) = (1 + a) x satisfies D exp(a · x) = a exp(a · x) and exp(a · (x + y)) = exp(a · x) exp(a · y). Define sin(a · x) and cos(a · x) as the real and imaginary part of exp(ia · x). From De a·x = ae a·x , we deduce D sin(a · x) = a cos(a · x) and D cos(a · x) = −a sin(a · x). Since exp is monotone, its inverse log is defined. Define the reciprocal X −1 by X −1 x = D log(x) and X −k = (X −1) k. We check identities like log(xy) = log(x) + log(y) from which log(1) = …