  We study the relationships between Gateaux, weak Hadamard and Frechet
differentiability and their bornologies for Lipschitz and for convex functions.
In particular, Frechet and weak Hadamard differentiabily coincide for all
Lipschitz functions if and only if the space is reflexive (an earlier paper of
the first two authors shows that these two notions of differentiability
coincide for continuous convex functions if and only if the space does not
contain a copy of $\ell_1$). We also examine when Gateaux and weak Hadamard
differentiability coincide for continuous convex functions. For instance,
spaces with the Dunford-Pettis (Schur) property can be characterized by the
coincidence of Gateaux and weak Hadamard (Frechet) differentiabilty for dual
norms.
