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let g(x) be coo and strictly increasing but nowhere real-analytic.

g(g(x)) = f(x) and f(x) is strictly increasing and real-analytic.

1) what are examples of such f(x) and g(x) ?

2) if f(x) = exp(x) , then what is g(x) ?

3) if f(x) = 2 sinh(x) , then what is g(x) ?

as you might expect , this relates to my sinh method ...