Sufficient condition for differentiability

Sufficient condition for differentiability: Let $U\subseteq\mathbb{R}^n$ be open and let $f:U\to\mathbb{R}^m$ be a function. Fix $a\in U$. Assume that each first-order partial derivative $\partial f_i/\partial x_j$ exists on a neighborhood of $a$ and is continuous at $a$ (for all components $i=1,\dots,m$ and coordinates $j=1,\dots,n$). Then $f$ is differentiable at $a$ in the sense of the Fréchet derivative, and its derivative is the linear map represented by the Jacobian matrix at $a$.

In particular, continuity of the first partial derivatives is a practical hypothesis for verifying differentiability, and it places $Df(a)$ in the framework of linear maps between Euclidean spaces.