In classical topology, homotopy theory depends fundamentally on continuous paths. Applying these standard concepts to discrete or granular spaces, such as rough sets, presents obvious mathematical challenges. Specifically, strict continuity in discrete environments often restricts admissible paths to constant maps, making the classical fundamental group trivial and largely uninformative for structural analysis. To address this issue within Pawlak approximation spaces (X, R), we propose a new algebraic invariant: the Rough Fundamental Group, denoted by π₁ᴿ(X, x₀ᴿ). Our approach replaces standard continuous paths with "rough paths," defined as finite sequences of transitions between adjacent granules, or equivalence classes. By equipping the time index with a discrete (granular) structure, we ensure both temporal and spatial stability, creating a functional discrete homotopy framework. We show that the resulting rough path homotopy classes form a well-defined group under the operation of rough concatenation. Additionally, we prove that rough continuous maps induce group homomorphisms, establishing the functorial nature of our construction. To test this invariant, we compute the rough fundamental group for a discrete model of the Rough Circle, proving it is isomorphic to the group of integers, ℤ. Conversely, the group is trivial for a contractible rough disk. These results confirm that π₁ᴿ(X, x₀ᴿ) successfully detects topological holes and analyzes connectivity in granular spaces.