Dear Wang Xiong,
You raise an interesting issue: differentiability in Nature. But whereas you resort to non-differentiable manifolds of Spacetime to correct existing mathematical models of the Universe, I argue in my essay, "The Metaphysics of Physics", all mathematical models of 'what is' the Universe are in essence metaphysical and will ultimately fail. Knowing truly 'what is' the Universe is like knowing another person; what they think, how they feel and how they will behave. We can only know ourselves truly. All knowledge in this sense is therefore self-knowledge.
The question of differentiability in Nature can be considered in the following very simple context, for example. When we take a thermometer reading, is the final reading on the thermometer "asymptotically approximated" (as would be the case if the underlying physical processes were differentiable) or will the final reading be exact (up to the limits of our instruments).
In my view, there is a simple explanation that does not require differentiability and is able to explain the reading we get would be exact and not asymptotic. The physical view here can be described as follows: "energy propagates continuously as a wave while it manifests discretely in interactions". Thus, we can view Nature as being 'continuous' and explain the 'manifestation' of Nature (as readings in our instruments, for example) as being 'discrete'.
Constantinos
