  Given a continuous real-valued function on [0, 1], and a closed subset E
\subset [0, 1] we denote by f E the restriction of f to E, that is, the
function defined only on E that takes the same values as f at every point of E
>. The restriction f E will typically be "better behaved" than f . It may have
bounded variation when f doesn't, it may have a better modulus of continuity
than f, it may be monotone when f is not, etc. All this clearly depends on f
and on E, and the questions that we discuss here are about the existence, for
every f, or every f in some class, of "substantial" sets E such that f E has
bounded total variation, is monotone, or satisfies a given modulus of
continuity. The notion of "substantial" that we use is that of either Hausdorff
or Minkowski dimensions.
