Why do derivations take only smooth inputs?

Question

In ‘An Introduction to Manifolds’ Tu introduces a concept of derivations at a point to define tangent spaces. The one thing I do not quite get is why derivations act only on smooth function. Why can’t the input be, say, only twice differentiate function? Similarly, when Zorich is talking about a tangent space to a manifold, he asks to prove that the tangent space built upon tensors is isomorphic to the tangent space built upon derivations for smooth manifolds. But why can’t we prove this for a manifold of any smoothness greater than 1?

Answer 1

They indeed can be. But this would ask you to keep track of a lot of differentiability levels. You can have $C^k$-manifolds for every $k$, and every operation happens at the appropriate $k$. By simply letting $k = \infty$, you can safely forget how many times you've differentiated, and have a much simpler theory.

Sometimes this is very important. For example, you cannot have a smooth embedding of the torus into $\mathbb R^3$ that is everywhere flat. But you can have one in $\mathbb R^4$: simply take $\{(x,y,z,w) \mid x^2+y^2 = z^2 + w^2 = 1\}$. But inspecting the proof, you'll see that it requires the manifold to be $C^2$, and "flat torus" still makes some certain sense in the $C^1$ realm. And in fact, you can have a flat embedding of a torus that is everywhere non-smooth, with a fractal-like structure.