What additional axiom to GARP do we need to generate a differentiable or smooth utility function After researching for a while, I find this:
https://www.jstor.org/stable/1913607?seq=2#metadata_info_tab_contents
They come up with an axiom called SSARP that generates a preference with smooth demand function. But what about smooth utility function? What axiom can guarantee that?

The axiom should be best on the demand data set just like GARP
None if the axiom is to be on preferences, as any smooth utility representation can be monotone transformed into a non-smooth utility function.