Es sei

${\displaystyle f\colon \mathbb {R} ^{d}\longrightarrow \mathbb {R} }$

eine messbare integrierbare Funktion. Zu einem fixierten Startpunkt ${\displaystyle {}(a_{1},\ldots ,a_{d})\in \mathbb {R} ^{d}}$ betrachten wir (für ${\displaystyle {}(x_{1},\ldots ,x_{d})\in \mathbb {R} _{\geq a_{1}}\times \cdots \times \mathbb {R} _{\geq a_{d}}}$) die Abbildung

${\displaystyle {}F(x_{1},\ldots ,x_{d}):=\int _{[a_{1},x_{1}]\times \cdots \times [a_{d},x_{d}]}fd\lambda ^{d}\,.}$

a) Sei ${\displaystyle {}f}$ stetig. Zeige

${\displaystyle {}{\frac {\partial }{\partial x_{1}}}{\frac {\partial }{\partial x_{2}}}\cdots {\frac {\partial }{\partial x_{d}}}F=f\,}$

${\displaystyle {}F(x_{1},\ldots ,x_{d})}$

${\displaystyle {}x\in \mathbb {R} ^{d}}$