Given a continuous function $$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ f : R → R , we denote the so-called “big Lip” and “little lip” functions by "Equation missing" and "Equation missing" respectively. We are interested in the following question. Given a set $$E \subset {\mathbb {R}}$$ E ⊂ R , is it possible to find a continuous function f such that "Equation missing" or "Equation missing"? For monotone continuous functions we provide a rather straightforward answer. For arbitrary continuous functions the answer is much more difficult to find. We introduce the concept of uniform density type (UDT) and show that if E is $$G_\delta $$ G δ and UDT then there exists a continuous function f satisfying "Equation missing", that is, E is a "Equation missing" set. In the other direction we show that every "Equation missing" set is $$G_\delta $$ G δ and weakly dense. We also show that the converse of this statement is not true, namely that there exist weakly dense $$G_{{\delta }}$$ G δ sets which are not "Equation missing". We say that a set $$E\subset \mathbb {R}$$ E ⊂ R is "Equation missing" if there is a continuous function f such that "Equation missing". We introduce the concept of strongly one-sided density and show that every "Equation missing" set is a strongly one-sided dense $$F_\sigma $$ F σ set.

We investigate doubling conditions defined in terms of measurable bounded sets and find a simple characterization of quasisymmetrically thick Cantor sets on the line.

AbstractWe denote the local “little” and “big” Lipschitz functions of a function f : ℝ → ℝ by lip f and Lip f. In this paper we continue our research concerning the following question. Given a set E ⊂ ℝ is it possible to find a continuous function f such that lip f = 1E or Lip f = 1E?In giving some partial answers to this question uniform density type (UDT) and strong uniform density type (SUDT) sets play an important role.In this paper we show that modulo sets of zero Lebesgue measure any measurable set coincides with a Lip 1 set.On the other hand, we prove that there exists a measurable SUDT set E such that for any Gδ set E͠ satisfying ∣EΔE͠∣ = 0 the set E͠ does not have UDT. Combining these two results we obtain that there exist Lip 1 sets not having UDT, that is, the converse of one of our earlier results does not hold.

Abstract. According to a classical result, if Ω is a domain in R d , where d > 1, f : Ω → R d is a homeomorphism and the lim-sup dilatation H f of f is finite almost everywhere on Ω, then f is differentiable almost everywhere on Ω. We show that this theorem fails if H f is replaced by the lim-inf dilatation h f . Our example demonstrates the sharpness of recent results of Kallunki and Koskela concerning the h f function and also of Balogh and Csörnyei involving the lower-scaled oscillation of continuous functions f : Ω → R.

Abstract. For each integer n ≥ 2 we construct a compact, geodesic metric space X which has topological dimension n, is Ahlfors n-regular, satisfies the Poincaré inequality, possesses R n as a unique tangent cone at Hn almost every point, but has no manifold points.