(Solution) - Show that f R a b with integral L if -(2025 Original AI-Free Solution)

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Show that f ? R*[a, b] with integral L if and only if for every ? > 0 there exists a gauge ge on [a, b] such that if
= {([xi-1, xi), ti)}ni=1 is any tagged partition such that 0 < xi - xi-1 > ??(ti) for i = 1, . . . , n, then |S(f, P) - L| < e. (This provides an alternate-but equivalent way of defining the generalized Riemann integral.)