a curve does not need to be injective? In diff. Geometry, curve is a differentiable mapping from an open interval to 3 dimensional euclidean space. Doesn't it need to be injective?
If it is not, then there might be a two different tangent vector at a point in the euclidean space...which make frenet analysis impossible at the point.
Any help will be appreciated!

You need to realise that a curve is the function $\gamma \colon [a,b]\to \mathbb R^3$, not its image. Even if the image looks like a figure eight (for instance), the function, if differentiable, will still have a tangent at every point, it's just that it will have a tangent at every point in $[a,b]$.
