The construction of tangent bundle on a $C^\infty$ manifold, as for example in the book of Warner, uses the existence of double derivatives. Of course the tangent space for a point is first constructed in case of an open set in a Euclidean space and then the whole setup is glued up. But then in the neighborhood of a point the second derivatives might be different for a different $C^1$-chart around the point. So I suppose the tangent space/tangent bundle construction is for $C^2$-manifolds. 


Question is:



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> Since $C^2$ is apparently the minimum condition for existence of tangent bundle, is the tangent bundle just $C^1$, or it is again $C^2$?
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What is the order of differentiability of the tangent bundle of a C^2- manifold?
The tangent bundle of a $C^1$ manifold exists: it's a $C^0$ manifold.  

Similarly, for every $n$, the tangent bundle of a $C^n$ manifold is a $C^{n-1}$ manifold.


