$f \in C(\mathbb R^n)$ be such that for some positive integer $m$ , $\Delta^m_{i,h}f=0 $ ; then $f$ is a polynomial in $n$ variables ? Let $f \in C(\mathbb R^n)$ be such that for some positive integer $m$ , $\Delta^m_{i,h}f=0 , \forall h \in \mathbb R , i=1,2,...,n$ ; then how to show that $f$ is a polynomial in $n$ variables ? Here $\Delta_{i,h}f(x)=f(x+he_i)-f(x),\forall x \in \mathbb R^n$ , where $e_i$ is the $i$-th standard basis vector 
 A: Fix $m \ge 2$ and denote by $V$ the space of continuous functions for which all the iterated differences $\Delta_{i,h}^m$ vanish. It is straightforward to see that $V$ is a linear subspace of $C(\mathbb{R}^n)$.
First fixing $h=1$, the fact that the $m$-th iterated differences $\Delta_{i,1}^m f$ vanish implies that for every fixed $x$, the values of $f(x + ke_i)$ for $k=1,\ldots,m$ determine $f(x)$. Similarly, for every fixed $x$, the values of $f(x-khe_i)$ for $k=1, \ldots, m$ determine $f(x)$. By induction, this implies that the values of $f$ on the integer lattice $\mathbb{Z}^n$ are determined by the values $f(x_1, \ldots, x_n)$ for integers $x_i \in \{ 1, \ldots, m \}$. It is easy to see that any function on these $m^n$ integer tuples can be interpolated by a polynomial whose degree in every variable separately is $\le m-1$. Also, any such polynomial belongs to the space $V$ since application of $\Delta_{i,h}^m$ decreases the degree in the $i$-th variable. This shows that the projection of $V$ onto the space of functions on the integer lattice (by restricting the domain) has dimension $m^n$.
Now the same argument shows that the projection of $V$ onto any dyadic lattice $2^{-k} \mathbb{Z}^n$ has the same dimension $m^n$, so the values of a function in $V$ on the integer lattice determine the values on any such dyadic lattice. Since dyadic lattices are dense and the functions are assumed to be continuous, the values on $\mathbb{Z}^n$ determine the functions in $V$, so $V$ has dimension $m^n$ and coincides with the space of polynomials whose degree in each variable is $\le m-1$.
