TITLE: Notion of smoothness for set-valued functions
QUESTION [8 upvotes]: Is there a way of talking about continuity and smoothness for set valued functions? More precisely, consider $M$ and $N$ topological/smooth manifolds, and let $f$ a function that associates to each point $p\in M$ a subset $f(p) \subset N$ (I haven't made any assumptions on what target sets are allowed, but feel free to discuss cases where some restrictions are required). Is there a meaningful/canonical way of saying that $f$ is continuous or smooth? 
For my particular application in mind, $M$ is a smooth manifold, and $f$ associates to each $p\in M$ an open, convex cone inside $T_pM$. 
Edit: I should clarify that a convex cone $K$ in some real vector space $V$ is a subset such that
(a) conic: for any $v\in K$ and $r\in \mathbb{R}_+$ $\implies rv \in K$. 
(b) convex: for any $v,w\in K$ and $a,b \in \mathbb{R}_+$ $\implies av + bw \in K$
It is open if $K$ is an open subset of $V$, so in particular open cones do not contain the origin.

REPLY [3 votes]: Three outcomes of a short brainstorming (all inspired by Algebraic Geometry):

I second Todd Trimble's comment to your question (it deserves to appear in an answer, so I repeat it here): Synthetic differential geometry gives you a way to talk about "manifolds of subsets", and you have sort of an automatic smoothness built in. But getting into this probably takes you on a long detour...
The Algebraic Geometer's way of treating many-valued functions is, very sloppily: Identify functions $X \rightarrow Y$ with their graphs, i.e. subvarieties $\Gamma \subseteq X \times Y$ with $pr_X(\Gamma)=X$ (defined on all of X) and $|pr_X^{-1}(x) \cap \Gamma|\leq 1 \ \forall x \in X$ (single-valued), then drop the second requirement.
So maybe you can look at smooth submanifolds of $M \times TM$ subject to the conditions you want?
The most reasonable, I would say, and close to the previous: Go to differentiable stacks. For your particular situation you could look at the classifying stack of cones inside tangent bundles: This would be the category fibred in groupoids over the site of differentiable manifolds which has objects $(M,C)$ with $M$ a smooth manifold, $C \subseteq TM$ a submanifold (smooth, except for the tip of the cone) of the total space of the tangent bundle such that each fiber is a cone in $T_x(M)$. Morphisms $(M,C) \rightarrow (M',C')$ should probably be smooth morphisms $f:M \rightarrow M'$ such that $C$ is the pullpack of $C'$ along $Tf$, the differential of $f$. Now a map of differentiable stacks from a manifold $N$ into this stack is the same as a smoothly varying choice of cones in the fibers of the tangent bundle $TN$. The same technique should work with other set-valued maps and is very flexible if you want to modify conditions.