On the range of the derivative of a real-valued function with bounded support
We study the set f 0 (X) = ff 0 (x) : x 2 Xg when f : X ! R is a dieren tiable bump. We rst prove that for any C 2 -smooth bump f : R 2 ! R the range of the derivative of f must be the closure of its interior. Next we show that if X is an innite-dimension al separable Banach space with a C p -smooth bump b : X ! R such thatkb (p) k1 is nite, then any connected open subset of X containing 0 is the range of the derivative of a C p -smooth bump. We also study the nite-dimensional case which is quite dieren t. Finally, we show that in innite-dimension al separable smooth Banach spaces, every analytic subset of X which satises a natural linkage condition is the range of the derivative of a C 1 -smooth bump. We then nd an analogue of this condition in the