Investigation on properties of continuous functions under variable-order fractional integral and its application in joint fire allocation modeling

As a natural extension of constant-order fractional operators, variable-order fractional calculus can characterize the spatiotemporal heterogeneity of systems in a more refined manner, yet its fundamental mathematical theory remains to be fully established. This paper is devoted to the regularity preservation problem of continuous functions under the variable-order Riemann-Liouville integral. It is rigorously proved that if the original function is continuous and of bounded variation, and the order function is continuously differentiable and satisfies an appropriate positive lower bound condition, then the function obtained after the variable-order integration remains continuous and of bounded variation. The proof, centered on the Jordan decomposition theorem, the Lebesgue dominated convergence theorem, and the integration-byparts estimation of total variation, provides a complete and rigorous argument. Furthermore, two typical functions of bounded variation are selected for numerical experiments. Under the action of different variable-order functions, the variable-order integrals are computed and the total variations are estimated. Numerical results demonstrate that the total variations of the integrated functions all remain finite, and the profiles exhibit a distinct smoothing effect, which intuitively validates the correctness of the theoretical conclusions. Meanwhile, partial conclusions are applied to military modeling to verify the effectiveness of the research findings. The work of this paper provides a rigorous mathematical foundation for the property preservation of variable-order fractional calculus on function spaces, and offers a reliable analytical framework for subsequent studies on topics such as Hölder continuity and fractal dimension evolution.