What is the Radius of Integrability in the Function Space F(R,R)?

This paper introduces the radius of integrability, a quantitative invariant that transforms the qualitative ϵ-δ formulation of Riemann integration into a measurable property of function spaces. For a Riemann integrable function and a prescribed accuracy ϵ, the radius identifies the largest partition mesh δ that guarantees every tagged Riemann sum approximates the integral within the specified error. The framework is developed for both compact domain intervals, via pointwise and uniform radii, and unbounded intervals, through the tail integrability radius which quantifies the necessary truncation window for improper integrals. Key theoretical results include the establishment of a bottleneck identity relating local and global mesh requirements and a structural theorem showing that for C1 integrands, the radius is asymptotically governed by the inverse of the function’s total variation. Furthermore, this work completes a hierarchical program of regularity radii—encompassing convergence, continuity, and differentiability—by revealing a dimensional progression of geometric anchors. We demonstrate that while continuity is anchored at a point and differentiability at a line, integrability is anchored at a two-dimensional region. The theory is illustrated through explicit computations for several classical functions, including the normal density, the stretched exponential, and the Thomae function, providing a new quantitative lens for classifying integrable functions based on their partition sensitivity and tail decay regimes.