Past exam of the short options course of the University of Oxford / 2017 / 4 / a

The map is differentiable at $x$ if there is a linear map $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ such thatEquivalently, $f(x+h)=f(x)+Lh+o(\Vert h\Vert )$. This linear map is unique and is the Fréchet derivative $D{f}_x$. Its matrix in the standard bases is the Jacobian matrix, whose $(i,j)$ entry is $\partial f_i/\partial x_j$ at $x$. A function is differentiable on its domain if this condition holds at every point.

A sufficient condition is that the derivative $D{f}_x:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is surjective, equivalently that its Jacobian matrix has rank $m$. Then the implicit function theorem makes the zero set near $x$ a $C^1$ embedded submanifold of dimensionThe rank condition persists nearby because the derivative is continuous. This is the local form of the regular level set theorem; only the rank at the specified point is needed for this local conclusion.