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On continuous functions with graphs of σ-finite linear measure

In this paper comparisons are made between the class of continuous functions of generalized bounded variation and the class of continuous functions with graphs having σ-finite length i.e. linear measure. An investigationof the differentiability and approximate differentiability of such functions discloses the fact that the latter class is considerably more extensivethan the former one. The following definitions will be needed: (1) A function f is said to be of bounded variation (VB) on a set E if where the supremum is taken over all sequences {[ai, bi]} of non-overlapping intervals with endpoints in E.