TWO LARGE SUBSETS OF A FUNCTIONAL SPACE

ABSTRACT. Let PI denote the Banach space composed of all bounded derivatives f of everywhere differentiable functions on [0,I] such that the set of points where f vanishes is dense in [0,i]. Let DO consist of those functions in PI that are unsigned on every interval, and let DI consist of those functions in PI that vanish on dense subsets of measure zero. Then DO and DI are dense G6-subsets of PI with void interior. Neither DO nor DI is a subset of the other. KEY WORDS AND PHRASES. Banach space of functions, derivative dense G-.b.at. 1980 MATHEMATICS SUBJECT CLSS9FICATION: 26A24 i. INTRODUCTION. The real vector space D of all bounded derivatives of everywhere differentiable functions on [0,i] is a Banach space [i] under the norm II f II sup If(x) l