Let $f \in \mathcal{H}(U)$. Show that if $f$ is $\mathcal{C}^\infty$ on $U$, then every partial derivative of any order of $f$ belongs to $\mathcal{H}(U)$.
Let $f \in \mathcal{H}(U)$. Show that if $f$ is $\mathcal{C}^\infty$ on $U$, then every partial derivative of any order of $f$ belongs to $\mathcal{H}(U)$.