How can I find the non-integer smoothness of a continuous and almost everywhere differentiable function?

Question

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?)

Motivation:

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.

REFERENCES:

• Holtz, O., Nazarov, F., Peres, Y., "New Coins from Old, Smoothly", Constructive Approximation 33 (2011).

Answer 1

In your specific example, the function is actually Lipschitz continuous. Such functions $f$ are $C^{\alpha}$ for each $0 < \alpha < 1$, and $\|f\|_{C^{\alpha}} \leq K_L$, where $K_L$ is any Lipschitz constant for $f$, for any $0 < \alpha < 1$.

Another example of Lipschitz continuous functions include the restrictions of concave or convex functions to $I=[0,1]$, when they are defined on an open interval containing $I$.

To determine in general a smoothness class, by differentiating the function, we can reduce to the case of a continuous non-$C^1$ function. Then, if the new function is Lipschitz continuous (e.g. differentiable almost everywhere with bounded derivative), the original function is $C^{d + \alpha}$ for each $\alpha < 1$, where $d$ is the number of continuous derivatives of the function.

Otherwise, you need to find a Hölder continuity index. This is harder, but if (for instance) there are finitely many non-$C^1$ points, you can consider the minimum of the Hölder indices in neighborhoods of such points. But it's not exactly the best way to bound the continuity modulus.