Funciones con derivadas sucesivas grandes y pequeñas por doquier

In this paper we show that, given two double sequences of positive real numbers, $\alpha$ and $\beta$, the subset of all functions defined on an open real set which have big derivatives and small ones with respect to $\alpha$ and $\beta$, at every point, is residual in $\mathcal{C}^\infty$. As a corollary, we derive that Baire-almost every function of $\mathcal{C}^\infty$ has null radius of convergence at each point.