Continuous partials of a given order implies differentiable that many times

Statement

Suppose $k$ and $n$ are positive integers.

Suppose $f$ is a real-valued function of $n$ variables $x_{1},x_{2},\dots ,x_{n}$. Suppose that $(a_{1},a_{2},\dots ,a_{n})$ is a point in the domain of $f$ such that all the $k^{th}$ order higher partial derivatives of $f$ exist and are continuous at and around $(a_{1},a_{2},\dots ,a_{n})$ (i.e., they exist and are continuous in an open ball containing $(a_{1},a_{2},\dots ,a_{n})$). Then, $f$ is $k$ times differentiable at $(a_{1},a_{2},\dots ,a_{n})$.

Here, by $k$ times differentiable, we mean that it is possible to iterate the Jacobian matrix operation $k$ times on $f$ at the point, where prior to every iteration, we flatten out the matrix we obtain as a vector. The first iteration gives the gradient vector, the second iteration gives the Hessian matrix, and the $k^{th}$ iteration in general gives something with $n^{k}$ coordinates.