This paper proposes and systematically elaborates a novel geometric framework—non-differential geometry—whose core lies in entirely abandoning the reliance on smoothness ($C^1$ or higher continuity) required by traditional differential geometry, demanding only $C^0$ continuity for geometric objects. By introducing new mathematical tools based on limits and infinite series, non-differential geometry overcomes the smoothness constraints in calculating geometric quantities such as curvature. Furthermore, this paper constructs a unique ``integration tool" (distinct from classical integration theory) specific to non-differential geometry, providing a novel approach for analyzing geometric objects. Current research focuses on Euclidean space, but the theoretical framework itself is not confined to any specific spatial structure and is applicable across low- to high-dimensional spaces. Non-differential geometry completely resolves the contradiction between continuity and differentiability and offers potential support for theoretical innovations in fields such as physics.