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 $T_pM$.

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.