= Critical value
{title2=$f(x_c)$}
A <critical point of a smooth map> is an input; a <critical value> is an output reached from at least one such input. For the fold $F(x,y)=(x,y^2)$, all inputs $(a,0)$ are <critical points of a smooth map>, and all outputs $(a,0)$ are <critical values>. The identical coordinate pairs live in different copies of $\mathbb R^2$. To classify an output $(a,b)$, inspect its entire <fiber of a function>:
$$
F^{-1}(a,b)=\begin{cases}\{(a,\sqrt b),(a,-\sqrt b)\}&b>0,\\\{(a,0)\}&b=0,\\\varnothing&b<0.\end{cases}
$$
For $b>0$ both inputs have <matrix rank> two, so the output is a <regular value>. For $b=0$ its input has <matrix rank> one, so the output is a <critical value>. For $b<0$ the <fiber of a function> is the <empty set>, and the output is a <regular value> despite being outside the <image of a function>.
In general, for a <smooth map> $f:X\to Y$, write $C_f$ for its set of <critical points of a smooth map>. Its set of <critical values> is $f(C_f)\subseteq Y$. Equivalently, $q\in Y$ is a <critical value> if some $p\in f^{-1}(q)$ has a <differential> that is not <surjective>. A <regular value> requires a <surjective> <differential> at every input in that <fiber of a function>. The universal condition holds vacuously for an empty <fiber of a function>: every output outside $f(X)$ is a <regular value>. These quantifiers are explicit in https://www.math.toronto.edu/~jeffrey/matd67/tu.pdf[Loring W. Tu, An Introduction to Manifolds, Definition 8.22 and its two following remarks].
\Image[/critical-points-values-fibers.png]
{title=Regular circular fibers and a critical value reached from both critical and regular inputs}
{id=critical-points-values-fibers}
The distinction is useful when solving equations. For a <smooth map> $f:X^m\to Y^n$ between <smooth manifolds> without boundary and a <regular value> $q$ with nonempty <fiber of a function>, the <regular level set theorem> makes $f^{-1}(q)$ an <embedded submanifold> of <dimension> $m-n$, with <tangent space>
$$
T_pf^{-1}(q)=\ker df_p.
$$
The <implicit function theorem> explains the freedom count: locally, $n$ independent equations fix $n$ coordinates and leave $m-n$ free. The <rank-nullity theorem> gives the same count for the displayed <kernel of a linear map>. See https://www.math.toronto.edu/mein/teaching/LectureNotes/Man.pdf[Eckhard Meinrenken, Differentiable Manifolds, Theorems 1.11 and 1.15] for the local projection description and its consequence for <level sets>.
For example, $H:\mathbb R^2\to\mathbb R$, $H(x,y)=x^2+y^2$, has <gradient> $(2x,2y)$ and just one <critical point>, the origin, with <critical value> zero. For $c>0$, its <fiber of a function> is the <circle> of <radius> $\sqrt c$, a one-dimensional <embedded submanifold>; negative $c$ are <regular values> with empty <fibers>. At $c=0$, the <fiber of a function> is a single point instead of a <circle>. A <critical value> does not force its <fiber of a function> to be nonsmooth: that singleton is itself a zero-dimensional <embedded submanifold>, but the regular <dimension> conclusion $2-1=1$ does not apply.
Nor does a single regular input certify a <regular value>. The scalar <smooth function> $h(t)=t^2(t-1)$ has $h^{-1}(0)=\{0,1\}$ and <derivative> $h'(t)=3t^2-2t$. The input zero is a <critical point>, while the input one is regular because $h'(1)=1$. Their shared output zero is nevertheless a <critical value>. Conversely, different <critical points of a smooth map> can share one <critical value>: $f(t)=(t^2-1)^2$ sends both critical inputs $-1$ and $1$ to zero. Counting inputs and counting critical outputs are different tasks.
Finally, for $G(t)=(t,t^2):\mathbb R\to\mathbb R^2$, every output on the <parabola> is a <critical value>, and every output off it is a <regular value> with empty <fiber of a function>. There is no nonempty regular <fiber of a function> here, so the negative expression $1-2$ never becomes a claimed <manifold> <dimension>. Both diagrams show exact mathematical examples, not measured data.
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