  We consider the gradient method with variable step size for minimizing
functions that are definable in o-minimal structures on the real field and
differentiable with locally Lipschitz gradients. We prove that global
convergence holds if continuous gradient trajectories are bounded, with the
minimum gradient norm vanishing at the rate $o(1/k)$ if the step sizes are
greater than a positive constant. If additionally the gradient is continuously
differentiable, all saddle points are strict, and the step sizes are constant,
then convergence to a local minimum holds almost surely over any bounded set of
initial points.
