Approximate differentiation: Jarńık points

We investigate Jarńık’s points for a real function f defined in R, i.e. points x for which lim apy→x |(f(y)− f(x))/(y − x)| = +∞. In 1970, Berman has proved that the set Jf of all Jarńık’s points for a path f of the one-dimensional Brownian motion is the whole R almost surely. We give a simple explicit construction of a continuous function f with Jf = R. The main result of our paper says that for a typical continuous function f on [0, 1] the set Jf is c-dense in [0, 1]. 0. Introduction and notation. For an arbitrary real function f of a real variable the set of points x where f ′ ap(x) = +∞ is of measure zero, and so is the set of x at which limy→x |(f(y)− f(x))/(y − x)| = +∞. Surprisingly, the “natural joint generalization” of these facts does not hold. In fact, Jarńık [4] has constructed a continuous function f such that lim apy→x |(f(y)− f(x))/(y − x)| = +∞ for almost all x and a function g of Baire class 2 such that lim apy→x |(g(y)− g(x))/(y − x)| = +∞ for each x. We shall say that x is a Jarńık point for f if lim ap y→x ∣∣∣∣f(y)− f(x) y − x ∣∣∣∣ = +∞ and the set of all Jarńık points for f will be denoted by Jf . Almost forty years after Jarńık’s article, in the theory of stochastic processes [1] (cf. [2]) it was proved that almost every path of the one-dimensional Brownian motion serves as an example of a continuous function such that Jf = R. In the first section of our paper we construct a continuous function with Jf = R as the sum of an explicitly defined trigonometric series. In fact, our example is slightly stronger, since we show that instead of the difference quotient |(f(y)− f(x))/(y − x)| it is possible to consider the quotient |f(y)− f(x)|/φ(|y − x|), where φ is any prescribed increasing continuous function on [0,+∞) with φ(0) = 0. 88 J. Malý and L. Zaj ı́ ček The above fact that for a typical Brownian motion path f we have Jf = R naturally motivates the following question: What can be said about Jf for a typical continuous function in C([0, 1]) ? A result of the same article [4] of Jarńık immediately gives that μJf = 0 for a typical function in C([0, 1]). Theorem 2 of [10] implies an improvement of this fact: Jf is σ-porous (and thus also a first category set) for a typical function in C([0, 1]). On the other hand, the main result of the present paper contained in Section 3 says that Jf is c-dense in [0, 1] for a typical function in C([0, 1]). The proof is not easy, it is similar to Saks’ [7] proof of the fact that a typical function in C([0, 1]) is not a Besicovitch function. It has, similarly to Saks’ proof, two basic ingredients: (a) The Banach–Mazur game method which was in fact essentially used by Saks (cf. note on p. 103 of [9]). (b) A basic lemma concerning general real functions. Our basic lemma is proved in Section 2. We believe that it is also of independent interest since it gives an alternative proof of Theorem 9.7 of [8] which has a connection with the theory of the Denjoy integral. In the following μ denotes the Lebesgue measure on the set R of all real numbers. We denote by C the Banach space C([0, 1]) equipped with the supremum norm. We say, as usual, that a typical function in C has a property if the family of functions which do not have this property is a first category set. If a 6= b are real numbers we denote by co(a, b) the convex hull of {a, b}, i.e. co(a, b) = [a, b] if a < b and co(a, b) = [b, a] if b < a. Recall that Jf = { x : lim ap y→x ∣∣∣∣f(y)− f(x) y − x ∣∣∣∣ =∞} . 1. A continuous function with Jf = R. Recall that the first proof of the existence of such a function is probably contained in [1], where it is shown that a typical Brownian path serves as an example. For our simple explicit construction we need the folowing easy lemma. Lemma 1. Let I be an interval of length p > 0, M ⊂ I and 0 < α < 1. Suppose that ∣∣∣∣ sin 2πx p − sin 2πy p ∣∣∣∣ ≤ α for every x, y ∈M . Then μ(M) ≤ 3p π arc cos(1− α) . Approximate differentiation: Jarńık points 89 P r o o f. Let J be an interval of the form [kp/4 , (k + 1)p/4], where k is an integer. Then clearly μ(M ∩ J) ≤ p 2π μ{x ∈ [0, π/2] : sinx ≥ 1− α} = p 2π arc cos(1− α) . Since I is contained in a union of 6 intervals of this type, we conclude that μ(M) ≤ 6 p 2π arc cos(1− α) . Theorem 1. Let φ : [0,∞) → [0,∞) be an increasing continuous function with φ(0) = 0. Then there exists a continuous function f on R such that (1) lim ap y→x |f(y)− f(x)| φ(|y − x|) =∞ for each x ∈ R . P r o o f. Put ψ = √ φ. It is easy to see that if (2) each z ∈ R is a density point for the set {x : |f(x)−f(z)| > ψ(|x−z|)}, then (1) holds. Now find a sequence (an)n=1 such that an > 0 and