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We construct a C-function f : [0, 1] → R such that for almost all x ∈ (0, 1), there is r > 0 for which f(y) > f(x) + f (x)(y − x) when y ∈ (x, x+r) and f(y) < f(x)+f (x)(y−x) when y ∈ (x−r, x). The existence of such functions is related to a problem concerning conical density properties of Hausdorff measures on Rn. We also discuss the tangential behavior of typical C-functions, using an improvement of Jarnik’s theorem on essential derived numbers.

Tangential Behavior of Functions and Conical Densities of Hausdorff Measures