This is a really niche post but..... I will hit you with it anyway:
Definitions. Let the instantaneous spiritual state be a function f: [0, T] → {0, 1} (flesh, Spirit), bounded, with at most countably many discontinuities (i.e., it can flip state, but not infinitely erratically in any interval — a reasonable real-world assumption). Let S(t) = ∫₀ᵗ f(τ) dτ be accumulated experience.
Theorem. S is continuous on [0, T], even though f itself may be discontinuous everywhere it flips.
Proof. f is Riemann integrable on [0, T] (bounded with discontinuities of measure zero, by the Riemann–Lebesgue criterion). For any Riemann-integrable f, S(t) = ∫₀ᵗ f is Lipschitz continuous with constant ≤ sup|f|, hence continuous — a direct consequence of the Fundamental Theorem of Calculus. ∎
Corollary. No felt continuity or gradualness in S can be used to infer that f itself is continuous or graded. The inference "my experience feels gradual, therefore my underlying state must be gradual" is invalid regardless of the specific f — this holds for any bounded f, binary or not.
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