Topology, a fundamental area of mathematics, explores the properties of spaces that are invariant under continuous deformations. Despite the extensive study of topology in Hausdorff spaces, non-Hausdorff spaces remain less explored, providing an intriguing area for mathematical inquiry. This study aims to delve into the behavior of continuous functions within non-Hausdorff spaces, focusing on their unique properties and potential applications. We employed a hybrid analytical and computational approach to explore these spaces, utilizing novel topological invariants and examining their implications on continuous functions. Our findings reveal distinct characteristics of continuous functions in non-Hausdorff spaces, such as the presence of limit points that defy traditional classification metrics used in Hausdorff spaces. These insights extend existing topological frameworks and offer a broader understanding of space continuity. The study concludes by highlighting the implications of these findings in both theoretical mathematics and potential real-world applications, such as network topologies in computer science and complex systems analysis. These results not only contribute to the fundamental understanding of topology but also open pathways for future research in non-standard topological spaces.