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

This paper introduces a quantitative refinement of the classical concept of differentiability within the space of real functions. Shifting the focus from the qualitative existence of the derivative to a scale-sensitive framework, we define two new invariants of the radii of differentiability: the radius of pointwise differentiability and the radius of uniform differentiability. These radii quantify the maximal horizontal scale over which the first-order Taylor approximation remains valid for a prescribed error tolerance ε. The theoretical development establishes a robust set of structural properties, including scaling laws, monotonicity, and behavior under function composition and sums. We provide a rigorous characterization of these invariants, demonstrating that the property of having an infinite radius of differentiability is uniquely characteristic of affine functions. A significant portion of the study is dedicated to the “bottleneck identity,” which reconciles local and global regularity by expressing the uniform radius as the infimum of its pointwise counterparts. Furthermore, we explore the interplay between these differentiability radii and the radius of continuity, utilizing the Fundamental Theorem of Calculus to prove that the process of integration yields a strict improvement in the local regularity profile. Finally, the utility of the proposed framework is demonstrated through explicit computations for several classes of elementary functions, including polynomials and trigonometric maps.