Local convergence of Newton method
It is well known that Newton method converges quadratically to the root of a function if the initial point is close enough to the root and the Hessian is Lipschitz continuous. x_{k+1} = x_k - \left[\nabla^2 f(x_k)\right]^{-1} \nabla f(x_k) If you will consider the following function: f(x) = \begin{cases} (x - 1)^2, & x \leq -1 \\ -\frac{1}{4}x^4 + \frac{5}{2}x^2 + \frac{7}{4}, & -1 < x < 1 \\ (x + 1)^2, & x \geq 1 \end{cases} You will see, that it is strongly convex function with smooth gradient (i.e. Lipschitz continuous Hessian).
