Abstract. In 1927, A. Khintchine proved that a measurable symmetrically differentiable function / mapping the real line R into itself is differentiable in the ordinary sense at each point of R except possibly for a set of Lebesgue measure zero. Here it is shown that this exceptional set is also of the first Baire category; even more, it is shown to be a o-porous set of E. P. Dolzenko.

Abstract. In this paper, an analogue of the Denjoy-Young-Saks theorem concerning the almost everywhere classification of the Dini dérivâtes of an arbitrary real function is established in both the case where the exceptional set is of first category and the case where it is o-porous. Examples are given to indicate the sharpness of these results.

We consider continuous functions f which map the open unit disk D into the Riemann sphere W. For a point ζ on the unit circle C, we say that χ is a chord at ζ if χ is a chord of C having one endpoint at ζ and that Δ is a Stolz angle at ζ if Δ is a Stolz angle with vertex ζ. Suppose S denotes either a chord at ζ, a Stolz angle at ζ, or D.

An example is given of a continuous real function that is of ordered harmonic bounded variation but not of harmonic bounded variation.

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