  We present an elementary proof of a general version of Montel's theorem in
several variables which is based on the use of tensor product polynomial
interpolation. We also prove a Montel-Popoviciu's type theorem for functions
$f:\mathbb{R}^d\to\mathbb{R}$ for $d>1$. Furthermore, our proof of this result
is also valid for the case $d=1$, differing in several points from Popoviciu's
original proof. Finally, we demonstrate that our results are optimal.
