I knew that, if function $f:A\to\mathbb{R}$ of class $C^2(A)$ in an open set $A\subset \mathbb{R}^n$ has a maximum, or respectively the minimum, in $x_0\in A$, then the Hessian matrix is positive semidefinite or respectively negative semidefinite. Analogously if the Hessian matrix of $f\in C^2(A)$ is definite positive, or respectively negative, then $f$ has a minimum, or respectively a maximum, in $x_0\in A$.
I read in Kolmogorov-Fomin's (p. 504 here) that if function $f(x_1,...,x_n)$ has a minimum in point $(x_1^0,...,x_n^0)$ then in this point $d^2f\ge 0$. (Analogously, if in point $(x_1^0,...,x_n^0)$ we have a maximum, then $df^2\le 0$) and that if in point $(x_1^0,...,x_n^0)$ we have that $df=0$ and $d^2f$ is definite positive then $f(x)$ has a minimum in this point (analogously a maximum if $d^2f<0$).
I think that the continuity of all second order derivatives is to be implicitly assumed in an open set containing $(x_1^0,...,x_n^0)$: is that so, or can we relax this assumption? I thank you very much for any answer confirming my thought or proving the contrary!