Take a function such as f($\lambda$) = min($\lambda$, c) for some $\lambda$ in $[0, 1]$ and some constant c in $(0, 1)$. This function is continuous and almost everywhere differentiable (it is not differentiable at c).
The smoothness of a function ($\alpha$) is a non-integer that depends on a derivative's modulus of continuity (or whether that modulus is of a certain order), which I find hard to determine for an arbitrary function, unlike the question of how many derivatives it has. If we restrict smoothness classes to integers, the function f above is obviously C0 and not C1. But in terms of non-integer smoothness, is the function any C$\alpha$ in between? In particular, if a function is continuous everywhere but not differentiable everywhere, is the modulus of continuity found in terms of the function itself to find what smoothness class it has? (And how can a function's non-integer smoothness class in general be found, besides counting its derivatives?)
This question is important to me because a method given in Holtz et al. (2011) relies on this non-integer notion of smoothness to describe a method to calculate polynomials that converge from above and below to a function, and I would like to apply this method.
- Holtz, O., Nazarov, F., Peres, Y., "New Coins from Old, Smoothly", Constructive Approximation 33 (2011).