  The Baillon-Haddad theorem establishes that the gradient of a convex and
continuously differentiable function defined in a Hilbert space is
$\beta$-Lipschitz if and only if it is $1/\beta$-cocoercive. In this paper, we
extend this theorem to G\^{a}teaux differentiable convex functions defined on
an open convex set of a Hilbert space. Finally, we give a characterization of
$C^{1,+}$ convex functions in terms of local cocoercitivity.
