What Is the Radius of Continuity in the Function Space F(R, R)?

The standard $varepsilon$–$delta$ definition of continuity is inherently quantitative, yet the precise dependence of the admissible radius $delta$ on the accuracy $varepsilon$ and the base point $x_0$ is rarely treated as an independent mathematical object. In this paper, we introduce the textit{radius of continuity} through two variants: the radius of pointwise continuity and the radius of uniform continuity, defined as explicit numerical invariants that capture the maximal symmetric neighborhood on which a real-valued function maintains a prescribed tolerance. We establish the fundamental structural properties of these radii, including their behavior under algebraic operations such as sums, products, and compositions, and demonstrate their inverse relationship to the classical modulus of continuity. Furthermore, we prove that the finiteness pattern of these radii characterizes constant versus non-constant functions. To illustrate the utility of this framework, we derive closed-form expressions for the pointwise radius of quadratic polynomials and the uniform radius of the normal probability density function. These examples highlight how the radius of continuity encodes geometric and probabilistic features, such as local curvature and global scale parameters. Ultimately, this perspective bridges the gap between real analysis and quantitative methods in metric geometry, offering a concrete measure of the stability of a function’s continuity.