From Wikipedia:

In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if $A$ is a closed subset of a Euclidean space, then it is possible to extend a given function off $A$ in such a way as to have prescribed derivatives at the points of $A$. It is a result of Hassler Whitney. A related result is due to McShane, hence it is sometimes called the McShane–Whitney extension theorem.

I need a stronger theorem which allows for both the set and the function to vary continuously, and for the resulting extended function to vary continuously.

Consider Euclidean space $\mathbb R^d$ (for any $d \ge 1$), and let $\mathcal K$ denote the metric space of compact subsets of $\mathbb R^d$, equipped with the Hausdorff metric. Select $\alpha \in \mathbb N$, and let $X = C^\alpha(\mathbb R^d, \mathbb R)$ be the Fréchet space of $C^\alpha$-smooth functions.

For any compact set $K \in \mathcal K$, let $X_K = C^\alpha(K, \mathbb R)$ be the Banach space of $C^\alpha$-smooth functions defined on $K$, and let $\rho_K : X \to X_K$ be the restriction map.

Define $Y_K$ to be the Fréchet space of functions which are real analytic on $\mathbb R^d - K$, and $C^\alpha$-smooth on $K$. Let $\iota_K : Y_K \to X$ be the inclusion map, which forgets that a function is analytic and just preserves its $C^\alpha$-smooth character.

The classical Whitney theorem implies that for each $K \in \mathcal K$, there is a function $w_K : X_K \to Y_K$ for which the following diagram commutes: $$\begin{array}[ccc] ~Y_K & \rightarrow^{\iota_K} & X \\ \uparrow {w_K} & & \downarrow {\rho_K} \\ X_K & \rightarrow_{1_{X_K}} & X_K \end{array}$$ where $1_{X_K}$ denotes the identity function on $X_K$. Equivalently, $1_{X_K} = \rho_K \circ \iota_K \circ w_K$. This means that we may extend a function from $K$ analytically, embed it into $X$, then restrict back to $K$, and we're left with the original function.

I would like to ensure not only that the Whitney map is continuous in the extending function, but also that the compact set may be allowed to vary.

Define the coproduct $\mathcal X = \coprod_{K \in \mathcal K} \{K\} \times X_K$, and refine the topology so that the projection map $\mathcal X \to \mathcal K$ is continuous. This way, $\mathcal X$ is kind of like a fibration over $\mathcal K$, except the fibers aren't necessarily homotopically equivalent. A point in $\mathcal X$ encodes both a compact set $K$ and a $C^\alpha$-smooth function defined on that set. (I think this is an admissible construction topologically. Tell me if this is not well-defined.)

Similarly, let $\mathcal Y = \coprod_{K \in \mathcal K} \{K\} \times Y_K$ encode the analytically extended functions, with continuous projection map $\mathcal Y \to \mathcal K$.

Consider the product space $\mathcal K \times X$, along with the inclusion map $\iota : \mathcal Y \to \mathcal K \times X$ and the restriction map $\rho : \mathcal K \times X \to \mathcal X$.

Now for the Whitney-type question. Does there exist a continuous map $w : \mathcal X \to \mathcal Y$ so that the following diagram commutes? $$\begin{array}[ccc] ~\mathcal Y & \rightarrow^{\iota} & \mathcal K \times X \\ \uparrow {w} & & \downarrow {\rho} \\ \mathcal X & \rightarrow_{1_{\mathcal X}} & \mathcal X \end{array}$$

notdo what you want? – Willie Wong Jun 21 '13 at 7:34