C^k function characterised by continuous derivatives

Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $k \ge 1$ be an integer. Say that $f$ is of class $C^k$ on $U$ when the following is true for each component $f_j$ and each $r \le k$. Each partial derivative $\partial^r f_j / \partial x_{i_r} \cdots \partial x_{i_1}$ of order $r$ exists on $U$ and is continuous on $U$. Then $f$ is of class $C^k$ on $U$ if and only if these two conditions hold.

1. For each $r$ with $0 \le r \le k$, the iterated derivative $D^r f$ exists on $U$ and is continuous on $U$.
2. For each $r$ with $0 \le r < k$, the map $D^r f$ is differentiable on $U$.