Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

=== Your proposal to “define the definite integral as the subtraction of the values of an antiderivative (existing because u(s)u(s)u(s) is continuous)” uses two classical facts: ===
# On R\mathbb{R}R, a continuous scalar function has an antiderivative (the classical indefinite integral / FTC existence part).
# Then the definite integral is F(t)−F(0)F(t)-F(0)F(t)−F(0).

Both facts rely on the usual topology and the classical limit operator. In your extended algebra EEE they are not automatic. To make the same argument valid you must supply (or prove) all of the following for EEE:

A. A clear notion of topology/continuity for s↦f(s)s\mapsto f(s)s↦f(s) when f(s)∈Ef(s)\in Ef(s)∈E. (E.g. a metric or a norm making EEE a topological vector space or Banach space.)

B. A guarantee that continuity implies existence of an antiderivative in EEE (this is a nontrivial property: it is exactly the FTC-existence statement transplanted into EEE). In classical analysis this follows from constructing F(x)=∫0xf(s) dsF(x)=\int_{0}^{x} f(s)\,dsF(x)=∫0xf(s)ds via limits of Riemann sums and showing it is differentiable; in EEE you must show the same construction makes sense and yields a derivative equal to fff.

C. A compatibility statement that the subtraction F(t)−F(0)F(t)-F(0)F(t)−F(0) yields a meaningful definite integral in EEE and that this agrees with any other acceptable definition (Riemann sums, limits, Bochner integral, etc.) wherever both exist.

D. If you want the definite integral to land back in the classical space (not just in EEE), you must additionally prove that the antiderivative FFF produced from a classical-continuous integrand is a classical function (this is the point your earlier draft omitted).

Without A–C your “antiderivative definition” is only a notation, not a justified construction.