Implicit Functions

The classical implicit function theorem is discussed in all analysis books. We are particularly fond of Spivak [1970, p. 40]. The history of the theorem, and many of its variations, is discussed in [Krantz and Parks [2013] and a comprenhensive modern treatment, using the tools of convex and variational analysis, is in Dontchev and Rockafellar [2014].

Suppose $f:\mathbb{R}^n\otimes\mathbb{R}^m\mapsto\mathbb{R}^m$ is continuously differentiable in an open set containing $(x,y)$ where $f(x,y)=0.$ Define the $m\times m$ matrix $$A(x,y)\mathop{=}\limits^{\Delta}\mathcal{D}_2f(x,y)$$ and suppose that $A(x,y)$ is non-singular. Then there is an open set $\mathcal{X}$ containing $x$ and an open set $\mathcal{Y}$ containing $y$ such that for every $x\in\mathcal{X}$ there is a unique $y(x)\in\mathcal{Y}$ with $f(x,y(x))=0.$

The function $y:\mathbb{R}^n\mapsto\mathbb{R}^m$ is differentiable. If we differentiate $f(x,y(x))=0$ we find $$\mathcal{D}_1f(x,y(x))+\mathcal{D}_2(x,f(x))\mathcal{D}y(x),$$ and thus $$\mathcal{D}y(x)=-[\mathcal{D}_2f(x,y(x))]^{-1}\mathcal{D}_1f(x,y(x)).$$

As an example consider the eigenvalue problem $$\begin{align*} A(x)y&=\lambda y,\\ y'y&=1 \end{align*}$$ where $A$ is a function of a real parameter $x$. Then $$\begin{bmatrix} A(x)-\lambda I&-x\\ x'&0 \end{bmatrix} \begin{bmatrix} \mathcal{D}y(x)\\ \mathcal{D}\lambda(x) \end{bmatrix} = \begin{bmatrix} -\mathcal{D}A(x)x\\ 0 \end{bmatrix},$$ which works out to $$\begin{align*} \mathcal{D}\lambda(x)&=y(x)'\mathcal{D}A(x)y(x),\\ \mathcal{D}y(x)&=-(A(x)-\lambda(x)I)^+\mathcal{D}A(x)y(x). \end{align*}$$