I am always curious about that whether there exists a class of function which seems that more smooth than the $C^{\infty}$ class, while it is far from $C^{\omega}$ analytic function . 


From my point of view, the symbol $\infty$ in $C^{\infty}$ means countably many.
Now the question is that whether we can regard the symbol $\omega$ in $C^{\omega}$ as order (uncountably many ) in continuum theory 


I had to admit that this may not be a standard question in MO. While, I still did expect some remarks or even answers for this question. 



Class of functions between $C^{\infty}$ and $C^{\omega}$
There are many different classes. See for example the following paper and the literature cited therein:


* Andreas Kriegl, Peter W. Michor, Armin Rainer: The Convenient Setting for Denjoy--Carleman Differentiable Mappings of Beurling and Roumieu Type. Revista Matemática Complutense (2015), doi:10.1007/s13163-014-0167-1, [(pdf)](http://www.mat.univie.ac.at/~michor/DC-convenient.pdf)


