The Peano curve and I-approximate differentiability


Krzysztof Ciesielski & Lee Larson

Real Anal. Exchange 17 (1991-92), 608-622

A function f:R-->R is density continuous (I-density continuous, deep-I-density continuous) at the point x if it is continuous at x when the density topology (I-density topology, deep-I-density topology) is used on both the domain and the range. It is known that the first coordinate of the classical Peano area-filling curve is nowhere approximately differentiable, even though it is continuous and density continuous. In this paper we generalize this result by proving that the same function is also I-density and deep-I-density continuous, even though it is nowhere I-approximately differentiable.

We also give an example of a bounded I-approximately continuous function that is not a derivative.

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Last modified April 29, 1999.