Inspired by the theory of analysis on fractals, we construct an example of a continuous, monotone function on an interval, which has vanishing derivatives on a dense set and infinite derivatives on another dense set. Although such examples could be constructed by classical means of probability and measure theory, this one is more elementary and emerges naturally as a byproduct of some new fractal constructions. | Turk J Math 30 (2006) , 211 – 220. ¨ ITAK ˙ c TUB A Fractal Example of a Continuous Monotone Function with Vanishing Derivatives on a Dense Set and Infinite Derivatives on Another Dense Set ¨ B¨ unyamin Demir, Vakıf Dzhafarov, S¸ahin Ko¸cak, Mehmet Ureyen Abstract Inspired by the theory of analysis on fractals, we construct an example of a continuous, monotone function on an interval, which has vanishing derivatives on a dense set and infinite derivatives on another dense set. Although such examples could be constructed by classical means of probability and measure theory, this one is more elementary and emerges naturally as a byproduct of some new fractal constructions. Key words and phrases: Sierpinki Gasket, harmonic function. 1. Introduction Fractal analysis has been a rising field in analysis in the last decade. One of the key features of this new theory has been the invention of Laplacians and harmonic functions on fractals [1]–[4]. The typical example is the Sierpinski-gasket (SG) and the by-now classical construction of (real-valued) harmonic functions on this fractal goes as follows: Let α0 , α1 , α2 be arbitrary real numbers and set F (pi ) = αi on vertices of SG. Then define F (q0 ) = (α0 + 2α1 + 2α2 )/5, F (q1 ) = (2α0 + α1 + 2α2 )/5, F (q2 ) = (2α0 + 2α1 + α2 )/5. (see Figure 1). Applying this procedure iteratively one gets a function on the set of junction points AMS Mathematics Subject Classification: 28A80, 26A48, 58E20 211 ˙ DZHAFAROV, KOC ¨ DEMIR, ¸ AK, UREYEN p0 q2 p1 q1 l1 q0 r1 p2 Figure 1. First stages of the SG. and extending this continuously to the whole of SG one obtains a so-called harmonic function on SG with very nice properties. The restrictions of this function to line-segments inside SG (for example to [p1 p2 ]) are worth understanding. This restriction is known to be monotone (or piece-wise monotone on two pieces) [1] and hence differentiable almost everywhere. But nothing is known more specificly about
