The Hessian being positive semidefinite at a point means it's positive semidefinite around the point?

Question

If the Hessian matrix of a $C^2$ function $f$ is positive semidefinite at a point $a$, can I conclude it is positive semidefinite in a neighborhood of that point?

I think the answer should be yes because all the second order partials are continuous, so as long as $x$ is close enough to $a$ the quadratic form associated to $Hf(x)$ should be sufficiently close to the quadratic form associated to $Hf(a)$, is this right? If yes, can someone provide a more rigorous approach, rather than this intuitive idea?

I'm interested in this because knowing that:

1. A $C^2$ function, defined in an open convex set, with semipositive definite Hessian in every point of the domain is convex.
2. Every critical point of a convex function is a global minimum.

I would like to conclude that if the Hessian of a $C^2$ function is positive semidefinite at a critical point, then there is a minimum in that point.

Answer 1

The Hessian matrix of $f(x,y)=x^3+y^3$ at $(0,0)$ is a simple counterexample.

By the continuity, you can only conclude that the Hessian matrix is positive (semi)definite in a neighborhood of the point where the Hessian matrix is positive definite.