User blog comment:LittlePeng9/Fast growing hierarchy of analytic functions/@comment-30754445-20170617192307/@comment-1605058-20170617193753

The thing about analytic functions is that they are very smooth. In fact, analyticity is often considered to be the strongest type of smoothness for real or complex valued functions (by the way, the advantage of complex functions in this context is that the functions are also "rigid" in a way, but that'd make for a different conversation).

On the other hand, at least using this post of yours, already the F function fails to be differentiable - for $$0<x<1$$ it is $$10^x$$ and for $$1<x<2$$ it is $$10^{10^{x-1}}$$. The former has derivative $$\log 10$$ at $$1$$, whereas the latter has derivative $$10(\log 10)^2$$.