Each number a in the segment (0, ½) produces a Cantor set, Ca, by putting b = 1 − 2a and recursively removing segments of relative length b from the centres of the interval [0, 1] and the intervals which are subsequently generated. The distribution function of the uniform probability measure on Ca is a Cantor function, fa. When a = 1/3 = b, Ca is the standard Cantor set, C, and fa is the standard Cantor function, f. The upper derivative of f is infinite at each point of C and the lower derivative of f is infinite at most points of C in the following sense: the Hausdorff dimension of C is ln(2)/ln(3) and the Hausdorff dimension of S = {x ∈ C: the lower derivative of f is finite at x} is [ln(2)/ln(3)]2. The derivative of f is zero off C, the derivative of f is infinite on C — S, and S is the set of non-differentiability points of f. Similar results are established in this paper for all Ca: the Hausdorff dimension of Ca is ln (2)/ln (1/a) and the Hausdorff dimension of Sa is [ln (2)/ln (1/a)]2. Removing k segments of relative length b and leaving k + 1 intervals of relative length a produces a Cantor set of dimension ln(k + l)/ln(1/a); the dimension of the set of non-differentiability points of the corresponding Cantor function is [ln (k + l)/ln (1/a)]2.

Abstract.The main purpose of this note is to verify that the Hausdorff dimension of the set of points N* at which the Cantor function is not differentiable is [ln(2)/ln(3)]2 . It is also shown that the image of N* under the Cantor function has Hausdorff dimension ln(2)/ln(3).Similar results follow for a standard class of Cantor sets of positive measure and their corresponding Cantor functions.The Hausdorff dimension of the set of points N* at which the Cantor function is not differentiable is [ln(2)/ln(3)]2.Chapter 1 in [5] provides a nice introduction to Hausdorff measure and dimension; references [5][6][7] pursue the topic. We begin our proof with some notation and discussion. Let C denote the Cantor set. Let N+ (N~) denote the set of points at which the Cantor function does not have a right side (left side) derivative, finite or infinite. Then N* = N+ U N~ U {t: t is an end point of C} denotes the nondifferentiability set of the Cantor function. Although we will assume familiarity with [4], where Eidswick characterized N*, some material is repeated for completeness.A number t in C has a ternary representation t = (tx, ... , tt, ...), where /, = 0 or 2.Let z(az) denote the position of the Azth zero in the ternary representation of t;(la) If teN+,then limsup{z(AJ + 1)/z(aj)} > ln(3)/ln(2); (lb) If limsup{z(AJ + 1)/z(az)} > ln(3)/ln(2), then t e N+ . Let md denote the ^-dimensional Hausdorff measure, and put r = ln(2)/ln(3).We will compute the Hausdorff dimension of N* by verifying, then mdN* > Kd > 0; Kd will be specified later for a sequence of d 's increasing to r2. Condition (A) will be verified for each d satisfying the inequalities 1 > d > r2 by constructing a set E (depending on d) which contains N* and satisfies the equation mdE = 0. To verify (B), we will consider a sequence {d"} of

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