Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set A we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph like, or central graph like micro tangent sets at all. Finally we show that at almost all points Takagi’s function is graph like, and Weierstrass’s nowhere differentiable function is central graph like.
Suppose F ⊂ [0, 1] is closed. Is it true that the typical (in the sense of Baire category) function in C 1 [0, 1] is one-to-one on F? If dimBF < 1/2 we show that the answer to this question is yes, though we construct an F with dimB F = 1/2 for which the answer is no. If Cα is the middle-α Cantor set we prove that the answer is yes if and only if dim(Cα) ≤ 1/2. There are F’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented.