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The points of nondifferentiability of the Cantor ternary function are characterized in terms of the spacing of 0's and 2's in ternary expansions. One consequence is an illustration of Morse's intermediate-value theorem for Dini derivates. 1. The characterization. Let 0 be the Cantor ternary function, T the Cantor ternary set (see, e.g., [3, p. 213]). In view of the fact that 0 does all of its "climbing" on T, a nowhere dense set of measure zero, and because of the symmetry of its construction, one might think that both onesided derivatives exist everywhere on [0, 1] as + Xo or 0. A type of intermediate-value theorem by A. Morse [2] shows that this is not the case. In fact [2, p. 130], for each A>0, {x: 0+(x)=2} has the power of the continuum. On the other hand, it is easy to show that the upper derivates are always + X or 0. Also [6, p. 278], since 0 satisfies Banach's (T1) condition, the set 0(N*) has measure zero where N* is the set of points at which 0 fails to have a derivative, finite or infinite. In other words, 0'(t)= + oo at most points t of T in the sense of measure (the measure on T induced by 0). Finally, it follows from Morse's work [2, p. 128] that 0(Z) is residual in [0, 1] where Z is the set of points at which the lower derivates vanish. Consequently, 0_(t)=0=O+(t) at most points t of T in the sense of category. The following theorem (i) characterizes the points t of T at which 0'(t)-= + oo, (ii) characterizes the points t of T at which 0-(t)=0=0+(t), (iii) leads to a concrete illustration of Morse's theorem, (iv) gives a clear picturre of the smallness of 0(N*), and (v) leads to some interesting questions of a more general nature. THEOREM 1. Let t be a point of T which is not an endpoint of a complementary interval, let z(n) denote the position of the nth zero in its ternary expansion, and let A=lim inf 3z(n)/2z(n+1). Then A

A characterization of the nondifferentiability set of the Cantor function