Roughly speaking a differentiable curve is a curve that is defined as being locally the image of an injective differentiable function \gamma \colon I \rightarrow X from an interval of the real numbers into a differentiable manifold , often \mathbb{R}^n.
More precisely, a differentiable curve is a subset of where every point of has a neighborhood such that C\cap U is diffeomorphic to an interval of the real numbers.