Riesz fractional integro-differentiation is recognized as an analytical framework that defines fractional powers of the Laplace operator through potentials and hypersingular integrals. Furthermore, hypersingular integrals are established as natural extensions of partial differential operators to fractional orders, thereby enabling the development of fractional calculus for multivariate functions based on these operator classes. This study is devoted to the generalization of Riesz fractional calculus to abstract metric measure spaces, where classical results are subsumed as particular instances. The central objective is to enhance the theoretical framework with new fundamental results on weighted continuity characterized by specific generalizaitons of the Hölder condition. Three principal contributions are presented. First, the local modulus of continuity is refined through a generalized definition compatible with advanced theoretical constructs, extending its applicability to broader function classes on metric measure spaces. Second, Zygmund-type estimates are derived for power-weighted functions via a novel analytical approach, resolving technical challenges introduced by power-law weights that are absent in unweighted settings. Third, boundedness theorems are established to characterize the behavior of potential-type operators and hypersingular integrals on power-weighted variable generalized Hölder spaces, thereby completing the analytical foundation of Riesz fractional calculus in these generalized settings. The scientific significance of this work lies in its advancement of fractional calculus as a discipline within function analysis and operator theory. Specifically, the developed theory rigorously formalizes continuity properties of functions on abstract metric measure spaces, offering an approach that unifies classical and generalized perspectives. These contributions are anticipated to hold substantial implications for applications in integral equations and mathematical modeling, where the derived continuity estimates and operator boundedness properties may be leveraged to analyze complex systems under non-standard regularity assumptions.