Implicit function theorem with no active variables The implicit function theorem states that, for open $U\subset\mathbb{R}^n$, a function $F: U \to \mathbb{R}^{n-k}$ which is continuously differentiable and provided the derivative of $F$ is onto, at a point $\vec{c}\in U$ s.t. $F(\vec{c})= 0$,  there exists a neighborhood of $\vec{c}$ in which $F = 0$ implicitly defines $n-k$ pivotal variables as a function $g$ of the $k$ nonpivotal variables. 
My question is, what happens if we let $k=0$?
Specifically, if $F: \mathbb{R}^1 \to \mathbb{R}^1$, could we apply the implicit function theorem locally at some $c$?
 A: There is a sense in which the implicit function theorem becomes the inverse function theorem when $\ k=0\ $, although, strictly speaking, it's really the extension of the implicit function theorem to a neighbourhood of $\ \vec{c}\ $ which becomes the inverse function theorem.
In the implicit function theorem, the columns of the derivative of $\ F\ $ at $\ \vec{c}\ $ corresponding to the pivot variables must be linearly independent, and since $\ F\ $ is continuously differentiable, this remains true on some open neighbourhood $\ V\ $ of $\ \vec{c}\ $.  And because the derivative of $\ F\ $ at any point of $\ V\ $ has full rank, $\ F(V)\ $ is also an open neighbourhood of $0$ in $\ \mathbb{R}^{n-k}\ $.  For notational convenience, at the cost of a minor loss of generality that is more apparent than real, I shall assume that the non-pivot variables in $\ x\in\mathbb{R}^n\ $ are the first $\ k\ $, $\ x_1,x_2,\dots,x_k\ $.  Then it can be shown that if $\ \vec{d}\in F(V)\ $, with $\ \vec{d}=F(\vec{b})\ $, there is a neighbourhood $\ W\ $ of $\ \left(b_1,b_2,\dots,b_k\right)\ $ and a function $\ G_{\vec{d}}:W\rightarrow \mathbb{R}^{n-k}\ $, such that
$$
F\left(x, G_{\vec{d}}(x)\right)-\vec{d}=0\ \ \text{ for }\ \ x\in W\ ,
$$
which is the implicit function theorem for the function $\ F-\vec{d}\ $ in a neighbourhood of $\ \vec{b}\ $.
When $\ k=0\ $, the variables $\ x\ $ will disappear, and we're left with the identity
$$
F\left(G_{\vec{d}}\right)= \vec{d}\ \ \text{ for }\ \ \vec{d}\in F(V)\ .
$$
That is, considered as a function of $\ \vec{d}\ $, $\ G_{\vec{d}}\ $ is the inverse of $\ F\ $, guaranteed to exist in a neighbourhood of $\ F\left(\vec{c}\right)=0\ $, because the derivative of $\ F\ $ at $\ \vec{c}\ $ is an invertible $\ n\times n\ $ matrix in this case.
