# Chaos​ does not always prevail

One of the first astonishments I faced as an undergraduate mathematics student was that chaos prevails. What do I mean by that? Well, there are more irrational numbers than rational numbers, there are more non-computable functions than computable, and, finally, there are more continuous functions devoid of a derivative than those which have one. In the light of a myriad of examples alike, one might be easily persuaded that most of the mathematical objects are ill-behaved.

In my final undergraduate years, I came to realize this is not the case. In particular, using robust measurements, one can show that the set of somewhere differentiable functions in $C[0,1]^k, k>1$ is so thick that any compact set can be shifted in such a way that it becomes a subset of somewhere differentiable functions. This is one of the results obtained in my joint work with Adam Kwela: Differentiability of continuous functions in terms of Haar-smallness. A set A is thick if we can drag any compact set K and drop it inside A‘s territory.

Thickness implies that the underlying set is not Haar-null nor Haar-meager. Yes, it is what it sounds like, a generalization of a null set and a meager set. If we restrict ourselves to the subsets of the real-line, these notions are equivalent.

• A null set is a set with a zero length, area, or volume.
• A meager set is a set that is a sum of a sequence of sets of which closure has an empty interior.

• An example of a meager set that is not null is a fat Cantor set.
• An example of a null set that is not meager is the complement of the sum of Cantor sets with measures $0,\frac{1}{2},\frac{2}{3},\frac{3}{4}$ and so on.

Consequences of the thickness of the set of continuous multi-dimensional somewhere differentiable functions are particularly interesting. In our work, we proved that Banach-Mazur theorem generalizes to higher dimensions. Therefore, two initially similar (or even equivalent) notions diverge as we pass to infinite dimensions, i.e., to the set of functions. The set of continuous multi-dimensional somewhere differentiable functions is, on the one hand, meager, and yet, on the other, it is not Haar-meager.

Another corollary that raises an eyebrow is a sharp contrast with the one-dimensional case. The space where only one lonely $x$ plays a role is small in every reasonable sense. Here, the set of nowheredifferentiables is meager, Haar-meager, Haar-null, and what not! Adding a little $y$-buddy changes everything and we are left with meagerness only.