On the range of the derivative of a smooth mapping between Banach spaces

We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ( X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f′ (X) =A . Whenever f is only assumed Gteaux differentiable, new phenomena appear: for instance,there exists a mapping f from 1 () into 2, which is bounded, Lipschitz-continuous, and so that for all x,y 1 (), if xy, then f′ (x) f′ (y) >1 .