My question is the following: *Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic?*


**EDIT.** According to a theorem of Serge N. Bernstein (*Sur les fonctions absolument monotones*, Acta Mathematica, 52 (1928) pp. 1–66) if $f\in C^\infty(a,b)$ and $f^{(n)}(x)\ge 0$, for all $x\in(a,b)$, then $f$ extends analytically in a ball centered at $(a,0)\in\mathbb C$ and radius $b-a$! 



Is a function with nowhere vanishing derivatives analytic?
If $f$ is $C^{\infty}$ every derivative is continuous, so the hypothesis on $f$ implies that each derivative $f^{(n)}$ has constant sign. Such functions were studied by S. Bernstein and called **regularly monotonic**. In particular he proved in 1926 that a regularly monotonic function is real analytic.


[This 1971 AMM article](http://alpha.math.uga.edu/%7Epete/Boas71.pdf) by R.P. Boas provides a proof, more history, and further results along these lines. See also [this 1975 PAMS article of J. McHugh](http://www.ams.org/journals/proc/1975-047-02/S0002-9939-1975-0354974-3/S0002-9939-1975-0354974-3.pdf).


