$C^1$ critical loci and values for Euclidean maps

Let $m\ge 0$ and $n\ge 1$, let $U\subseteq {\mathbb{R}}^m$ be open, and let $F\in C^1(U,{\mathbb{R}}^n)$. Here $C^1$ has the componentwise Euclidean meaning of $C^r$ and smooth maps between smooth manifolds. For each $x\in U$, $D{F}_x$ is the total derivative: the linear map $L$ satisfying $F(x+h)=F(x)+Lh+o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ as in The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder. The $C^1$ critical locus of $F$ is

$${\mathrm{Crit}}_{C^1}(F):=\{x\in U:D{F}_x({\mathbb{R}}^m)\mathrm{\ne }{\mathbb{R}}^n\}\mathrm{.}$$

The $C^1$ critical-value set is

$${\mathrm{CV}}_{C^1}(F):=F({\mathrm{Crit}}_{C^1}(F))\mathrm{.}$$

A value outside this set is a regular value; in particular, a value with empty fibre is regular. This definition uses only the first derivative, so it applies to $C^1$ maps that need not be smooth.

When $m=0$, the only open domains are $\mathrm{\varnothing }$ and $\{0\}$. On $\mathrm{\varnothing }$ both sets are empty. On $\{0\}$, the unique total derivative ${\mathbb{R}}^0\to {\mathbb{R}}^n$ is zero and is not onto because $n\ge 1$, so the domain point is critical and its image is the critical-value set.