The Lipschitz Regularity and Differentiability of The Riemann-Liouville Fractional Integral of Holder Continuous Functions

This paper establishes regularity results for the Riemann-Liouville fractional integral applied to Hölder continuous functions. It has been proven that for a function [Formula: see text] in the Hölder space [Formula: see text] with [Formula: see text], its Riemann-Liouville fractional integral of order [Formula: see text] becomes Lipschitz continuous when [Formula: see text]. A complete proof has been provided covering all cases [Formula: see text] and [Formula: see text], with explicit piecewise estimates for the Lipschitz constants. Furthermore, conditions under which the fractional integral becomes differentiable are investigated. Numerical results with the Weierstrass function confirm the theoretical predictions and illustrate the smoothing effect of the fractional integration. The results provide a complete understanding of the regularity gain under the Riemann-Liouville fractional integration, which is fundamental for applications in fractional differential equations, signal processing and fractal analysis.