# not sure why this is smooth

#### slevvio

Define $$\displaystyle g: V \rightarrow \mathbb{R}$$ smooth where $$\displaystyle V \subseteq \mathbb{R}^n$$ is open. Then define a map, for a fixed $$\displaystyle x \in V$$

$$\displaystyle h_i(x) = \int_0^1 \frac{\partial g}{\partial x_i}(tx) dt$$

(we may assume tx is in V)

Can anyone explain to me why this function is smooth?

Thanks for any help

#### xxp9

$$\displaystyle \frac{\partial{h_i}}{\partial{x^j}} = \int_0^1 t\frac{\partial^2{g}}{\partial{x^ix^j}}(t \mathbf{x}) dt$$

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#### slevvio

hmmm why is this the case?