In their book "An introduction to Optimization", 4th Edition, Chong and Zak has following text. What does the $C^2$ mean?
Theorem 6.2 Second-Order Necessary Condition (SONC). Let $\Omega \subset R^n$, $f \in C^2$ a function on $\Omega$, $x^*$ a local minimizer of $f$ over $\Omega$, and $d$ a feasible direction at $x^*$. If $d^T \nabla f(x^*)=0$, then $d^TF(x^*)d \geq 0$, where $F$ is the Hessian of $f$.
It means $f\in C^2(\Omega)$, id est, that $f$ is differentiale twice at all points of $\Omega$, and that moreover its Hessian is continuous.