On the Dirichlet Continuity of Functions

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This paper, which is a supplement to our recent work [5] on the powerwise integration, consists of three mutually independent sections, each being concerned, respectively, with elucidating a doubtful point contained explicitly or implicitly in the paper [5]. As defined in [5], a function is called Dirichlet continuous on a compact nonconnected set Q, if it is continuous on Q and if it fulfils the Dirichlet condition on every compact nonconnected set contained in Q. Now the definition of the Dirichlet condition consists of three items. We are interested in examining whether or not the second item is superfluous. The answer is in the negative, as will be shown in § 1. As defined in [5], a function is said to fulfil the condition (P) on a linear set E, if either the function is AC on E, or else if there exists a CT null set which contains E and on which the function is Dirichlet continuous. It is the object of § 2 to show that the Dirichlet continuity cannot be replaced here by the Dirichlet condition, in the sense that the theory of the powerwise integration would collapse if we did so. We thus find that the Dirichlet continuity is essentially stronger than the Dirichlet condition. We proposed in [5] the following problem : To decide whether a function which is Dirichlet continuous on a compact nonconnected set, is necessarily powerwise continuous on this set. This problem will be solved in the negative in § 3. As in our previous papers, a function, by itself, will always mean a mapping of the real line R into itself, unless another meaning is obvious from the context. We shall also continue denoting by N the set of the positive integers and by M that of the nonnegative integers.
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Iseki Kanesiroo
Department Of Mathematics Faculty Of Science Ochanomizu University

On the Dirichlet Continuity of Functions

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