A constructive examination of rectifiability

Abstract

We present a Brouwerian example showing that the classical statement 'every
Lipschitz mapping f from [0; 1] to [0; 1] has rectiable graph' is essentially non-
constructive. We turn this Brouwerian example into an explicit recursive ex-
ample of a Lipschitz function on [0; 1] that is not rectiable. Then
we deal with the connections, if any, between the properties of rectiability
and having a variation: we show that the former implies the latter, but the
statement every continuous, real-valued function on [0; 1] that has a variation
is rectiableis essentially nonconstructive.