  We derive a simple criterion that ensures uniqueness, Lipschitz stability and
global convergence of Newton's method for the finite dimensional zero-finding
problem of a continuously differentiable, pointwise convex and monotonic
function. Our criterion merely requires to evaluate the directional derivative
of the forward function at finitely many evaluation points and for finitely
many directions.
  We then demonstrate that this result can be used to prove uniqueness,
stability and global convergence for an inverse coefficient problem with
finitely many measurements. We consider the problem of determining an unknown
inverse Robin transmission coefficient in an elliptic PDE. Using a relation to
monotonicity and localized potentials techniques, we show that a
piecewise-constant coefficient on an a-priori known partition with a-priori
known bounds is uniquely determined by finitely many boundary measurements and
that it can be uniquely and stably reconstructed by a globally convergent
Newton iteration. We derive a constructive method to identify these boundary
measurements, calculate the stability constant and give a numerical example.
