Consider the autonomous system for r E R": i(t) = f(x(t))

Let r>0 be some fixed constant, and let D = {r: xSr}. =

(1)

Suppose there exists a continuously differentiable function V, defined on D such that Vi(0) = 0 and V is positive definite along solutions of Equation (1). Furthermore, suppose for anyeE (0. r), Vi itself is not negative semidefinite on {r: |<e},

Prove that the origin is unstable.​