User blog comment:LittlePeng9/Fast growing hierarchy of analytic functions/@comment-11227630-20180105032538

This analytic FGH still has some problems.

In normal FGH, $$f_\alpha(1)=2$$, and (for many fundamental sequences) n = 2 is the minimal requirement to get large numbers in $$f_\alpha(n)$$.

But in your definition of analytic FGH, this "minimal requirement to get large numbers in $$f_\alpha(x)$$" of x doesn't exist, which means that for all positive real x, there is always some ordinal $$\alpha$$ such that $$f_\alpha(x)<x$$, i.e. doesn't "get large". So we can't build a FGH number series like this.