Most CĮ Functions Are Nowhere Gevrey Differentiable of Any Order

Abstract

We define a complete metric on $C^\infty$, and find that most functions in $C^\infty$ are nowhere Gevrey differentiable of any order. For any $s > 1$ we prove there exists an everywhere Gevrey differentiable function of order $s$ that is nowhere Gevrey differentiable of any order less than $s$.