Let $F: R^n \to R^m $ be a vector valued function with components $F = (f_1 ,f_2 ,..., f_m).$ What is the most proper and universal accepted definition for the function $F$ be twice differentiable at the point $x$? Does that simply mean $f_i$ be twice-differentiable at point $x$ for all $i = 1,2,...,m$. I know this is the case when we are talking about a function being $C^2$ (see What does it mean for a vector function to be twice differentiable? ).
Or does it mean $F$ be differentiable around $x$ and the derivative of $F$ which is a matrix valued function, denoted by $\nabla F : U \to R^m \times R^n $ be differentiable (in some sense) at the point $x$.
Any reference about that would be appreciated .