 Solution

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.