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. 



Notion of smoothness for set-valued functions
My idea is that if we want to compare $f(p)$ and $f(q)$ for nearby points $p$ and $q$, then we need to be able to put $f(p)$ and $f(q)$ into the same space. To do this, I'm going to assume that $M$ is a finite-dimensional Riemannian manifold, so that we can make use of a connection $\nabla$ on $M$.


For all $p$, let $f(p)$ be an open cone of $T\_p M$. Let $U\_p M$ denote the unit sphere in $T\_p M$, and define $$g(p) = f(p) \cap U\_p M.$$ Since $f(p)$ is a cone, it is the linear span of $g(p)$. Thus, any smoothness on $g$ will apply to $f$ as well.


Let $w \in T\_p M$, and define the covariant derivative of $g(p)$ in the direction $w$ by $$\nabla\_w g(p) := \{ \nabla\_w v \}\_{v \in g(p)}.$$


Let $\gamma$ be a smooth curve on $M$ with $\gamma(0) = p$ and $\dot \gamma(0) = w$. Define the parallel transport of $g(p)$ along $\gamma(t)$ by $$\nabla\_{\dot\gamma(t)} g(p) := \{ \nabla\_{\dot\gamma(t)} v \}\_{v \in g(p)}.$$ That is, the parallel transport of the set $g(p)$ is given by transporting each vector in $g(p)$ along the curve $\gamma(t)$.


Now, both $\nabla\_{\dot\gamma(t)} g(p)$ and $g(\gamma(t))$ are open subsets of the unit tangent space $U\_{\gamma(t)} M$ at the point $\gamma(t)$. If the set-valued function $g$ is to be smooth, then these two sets should be comparable. 


Let $\operatorname{Vol}\_q$ denote the (finite) volume measure on the unit sphere $U\_q M$ at the point $q \in M$, and let $\Delta$ denote the symmetric difference of two sets. Let us say that $g$ is smooth at $p$ in the direction $w$ if $$\operatorname{Vol}\_{\gamma(t)} \left( \nabla\_{\dot\gamma(t)} g(p) ~\Delta~ g(\gamma(t)) \right) = O(t)$$
for all smooth functions $\gamma$ with $\gamma(0) = p$ and $\dot \gamma(0) = w$. If $g$ is smooth at all points in all directions, then we shall say it is smooth on $M$. Consequently, we shall say that $f$ is smooth if $g$ is smooth.


