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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 $f$.

O. Altun
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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.

  • Do you have a name for that, so that I can look up on internet? Also, does it mean Gradient is also continuous? – O. Altun Oct 17 '18 at 14:04
  • Some say twice continuously differentiable function, but most people I know write something like $f$ is (in) $C^2$, or $f$ is a $C^2$ function, and read it C-two. Operatively, it just means that all the second partial derivatives exist at all points and that they are continuous functions. Yes, it implies that the gradient is a continuous map, and it also implies the non-obvious fact $\frac{\partial }{\partial x_i}\left[\frac{\partial f}{\partial x_j}\right]=\frac{\partial }{\partial x_j}\left[\frac{\partial f}{\partial x_i}\right]$ for all $i,j$. –  Oct 17 '18 at 14:31