There is a theorem due to Bernstein related to analytic functions :
If $f : ]0,1[ \to \mathbb{R}$ is an absolutely monotonous function (that is a $\mathcal{C}^\infty$ function such that for every $n$, we have $f^{(n)} \geqslant 0$) then $f$ is analytic.
The proof is a basic exercise in real analysis, but what are the applications ? The only applications I know are proving that $\exp$ or $\tan$ are analytic, but it would be ridiculous to prove this using the fact they're "absolutely continuous". If someone has precisions, examples or further information about it, let me know.