  Transitions between topologically distinct electronic states have been
predicted in different classes of materials and observed in some. A major goal
is the identification of measurable properties that directly expose the
topological nature of such transitions. Here we focus on the giant-Rashba
material bismuth tellurium iodine (BiTeI) which exhibits a pressure-driven
phase transition between topological and trivial insulators in
three-dimensions. We demonstrate that this transition, which proceeds through
an intermediate Weyl semi-metallic state, is accompanied by a giant enhancement
of the Berry curvature dipole which can be probed in transport and
optoelectronic experiments. From first-principles calculations, we show that
the Berrry-dipole --a vector along the polar axis of this material-- has
opposite orientations in the trivial and topological insulating phases and
peaks at the insulator-to-Weyl critical points, at which the nonlinear Hall
conductivity can increase by over two orders of magnitude.
