What does this mean:
"Suppose that the set of the derivatives: $\{(Df)_x\in\mathbb{L}(\mathbb{R}^n,\mathbb{R}^m):x\in[p,q]\}$ is convex".
I know the usual meaning of a convex set, but in this set, what is the precise definition?
What does this mean:
"Suppose that the set of the derivatives: $\{(Df)_x\in\mathbb{L}(\mathbb{R}^n,\mathbb{R}^m):x\in[p,q]\}$ is convex".
I know the usual meaning of a convex set, but in this set, what is the precise definition?
There is nothing really special about this case at all, provided you know the precise definition of a convex set in a (real) vector space. Namely, $C \subseteq V$ is convex if for every $x,y \in C$, and every $t \in (0,1)$ we have $tx + (1-t)y \in C$ also; in other words, given two points in the set $C$, the straight line segment between them is completely contained in $C$. In this case, we have $C = \{ (Df)_x \in \mathbb{L}(\mathbb{R}^n, \mathbb{R}^m) \,|\, x \in [p,q]\},$ so the statement in question simply means that given two points in $C$, say $(Df)_x$ and $(Df)_y$ for $x,y \in [p,q]$ and $t \in (0,1)$, there exists a $z \in [p,q]$ such that $(Df)_z = t(Df)_x + (1-t)(Df)_y.$ (Note that the same $z$ doesn't need to work for every $t$, and probably will not).