2019/10/31 by Zindulka, Ondřej · 1 citation
#26A16 #26A27 #26B05 #26B35 #28A78 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1910.14527
For a mapping f\colon X→ Y between metric spaces the function lip f\colon X→[0,∞] defined by lip f(x)=\liminfr→0(diam f(B(x,r)))/(r) is termed the lower scaled oscillation or little lip function. We prove that, given any positive integer d and a locally compact set Ω⊆ℝd with a nonempty interior, for a typical continuous function f\colon Ω→ℝ the set \x∈Ω:lip f(x)>0\ has both Hausdorff and lower packing dimensions exactly d-1, while the set \x∈Ω:lip f(x)=∞\ has non-σ finite (d-1)-dimensional Hausdorff measure. This sharp result roofs previous results of Balogh and Csörnyei, Hanson and Buczolich, Hanson, Rmoutil and Zürcher. It follows, e.g., that a graph of a typical function f∈ C(Ω) is microscopic, and for a typical function f\colon[0,1]→[0,1] there are sets A,B⊆[0,1] of lower packing and Hausdorff dimension zero, respectively, such that the graph of f is contained in the set A×[0,1]∪[0,1]× B.