 Critical set

For a differentiable map $F:M^n\to N^m$ between smooth manifolds, its critical set isHere critical means that the derivative is not onto the target tangent space, as in the preimage theorem. This convention matters when the dimensions differ. For a scalar-valued function, it is the set where the differential vanishes; in Euclidean coordinates it is the zero set of the gradient.

For $f(x,y)=x^2$, the critical set is the whole line $x=0$, so it need not consist of isolated critical points. An immersion such as $t\mapsto (t,0)$ from $\mathbb{R}$ to ${\mathbb{R}}^2$ has injective derivative at every point but every point is critical in the onto-target convention. Saying merely that full rank fails can obscure this distinction between immersion and submersion.