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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}^m$, and let $a \\in U$.\nSuppose that each [[partial derivative|09EGC6F8PT]] $\\partial f_j / \\partial x_i$ 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 $a$.\nThen $f$ is [[differentiable|0135XZZQ8M]] at $a$.\n\n## Remarks\n\nIf 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 $U$.\nThe 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 $0$.\nWithout continuity, the existence of the partial derivatives is not enough, see [[this counterexample|0TNZC55X3J]].\n\n## Explanations\n\n### L1 human\n\nThe partial derivatives give a candidate for the derivative: the matrix of all $\\partial f_j / \\partial x_i$ at $a$.\nContinuity of the partial derivatives near $a$ is what makes this candidate a good linear approximation in all directions, and not only along the axes.\nIn practice, this theorem is how you check differentiability: compute the partial derivatives and see that they are continuous.\n",
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