Let $f$ be a function defined on $(-\infty, a]$ such that every derivative of $f$ is strictly monotonic. Does it guarantee uniqueness of a smooth continuation $g$ of $f$ to the whole real line, where every derivative of $g$ is strictly monotonic? If not, what condition should we use so that the continuation is unique but other than requiring $f$ to be analytic?
EDITED. The following theorem of Bernstein answers the question:
If $f$ is infinitely differentiable on an interval and no derivative changes sign, then $f$ is analytic.
Your condition that all derivatives are monotone of course implies that none of them changes sign. Therefore, if such a function is extended on a larger interval with preservation of the property that no derivative changes sign, then such an extension is unique.
S. Bernstein, Sur la définition et les propriétés des fonctions analytiques d'une variable réelle, Math. Ann. vol. 75 (1914) pp. 449-468.
Here is a link: http://www.digizeitschriften.de/dms/img/?PID=GDZPPN00226580X The theorem is stated in section 5. Actually it is much stronger than I stated.
A survey of the later results on the topic is
Polya, G. On the zeros of the derivatives of a function and its analytic character. Bull. Amer. Math. Soc. 49, (1943). 178–191.