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.