Define g: V \rightarrow \mathbb{R} smooth where V \subseteq \mathbb{R}^n is open. Then define a map, for a fixed x \in V h_i(x) = \int_0^1 \frac{\partial g}{\partial x_i}(tx) dt (we may assume tx is in V) Can anyone explain to me why this function is smooth? Thanks for any help