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.

**LaTeX 2e source file**
and three postscript picture files:
PeanoIterate.ps,
PeanoIterate2.ps, and
PeanoX.ps.

**Requires rae.cls file**,
amsmath.sty, amssymb.sty, and epsf.sty.

**Last modified April 29, 1999.**