Continuous partial derivatives imply differentiability

Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$. Suppose that each partial derivative $\partial f_j / \partial x_i$ exists at each point of $U$, for $1 \le i \le n$ and $1 \le j \le m$, and that each of them is continuous at $a$. Then $f$ is differentiable at $a$.

Remarks

If the partial derivatives exist on $U$ and are continuous on all of $U$, then $f$ is differentiable on $U$, and $Df$ is continuous. So $f$ is of class $C^1$ on $U$. The converse is false. The function $f(x) = \|x\|^2 \sin(1/\|x\|)$ for $x \ne 0$, with $f(0) = 0$, is differentiable on $\mathbb{R}^n$, but its partial derivatives are not continuous at $0$. Without continuity, the existence of the partial derivatives is not enough, see this counterexample.

Explanations

L1 human

The partial derivatives give a candidate for the derivative: the matrix of all $\partial f_j / \partial x_i$ at $a$. Continuity of the partial derivatives near $a$ is what makes this candidate a good linear approximation in all directions, and not only along the axes. In practice, this theorem is how you check differentiability: compute the partial derivatives and see that they are continuous.