Restrictions of continuous functions

Given a continuous (or just measurable) real-valued function on [0, 1] and a closed subset E ⊂ [0, 1], denote by f|_E the restriction of f to E. f|_E can be "better behaved" than f and we discuss the existence, for every f, or every f in some class, of sets E that are substantial in terms of fractal dimension, such that f|_E has bounded total variation, or is monotone, or satisfies a given modulus of continuity.