Can continuous correspondence be represented via continuous functions?

Question

Let $\Theta \subset \mathbb{R}^n, \mathcal{X} \subset \mathbb{R^m}$, and suppose that $C: \Theta \rightrightarrows \mathcal{X}$ is a correspondence defined by $f: \Theta \times \mathcal{X}\to \mathbb{R}^d$ as follows:

$$ C(\theta) = \{ x\in \mathcal{X} \mid f_1(\theta, x) \geq 0, \dots, f_d(\theta, x) \geq 0 \} $$

Is it true that 1) if $f$ is continuous then $C$ also is, and 2) if $C$ is continuous then $f$ also is?

(Here continuity for correspondences is defined as upper and lower hemicontinuity.)

What does $f(\theta,x)\ge 0$ mean for $f(\theta,x)\in\mathbb{R}^d$? — Ben McKay, Feb 24 '23 at 17:25
What is $\theta$? If $\theta\in\Theta$, then what does $f_1(\theta,x)$ mean, given that $f$ is defined on $\Theta$? — Iosif Pinelis, Feb 24 '23 at 17:42
What topology do you have on the set of all subsets of $\mathcal X$? — Iosif Pinelis, Feb 24 '23 at 17:58
What is that "Euclidean" topology on the set of all subsets of $\mathcal X$? Can you describe it formally? — Iosif Pinelis, Feb 24 '23 at 18:02
That should be the discrete topology which is given as the collection of all subsets of $\mathcal{X}$. — Ded, Feb 24 '23 at 18:16
The discrete topology (which is of course not Euclidean in any sense) would be quite bad, as almost no map from a set with the Euclidean topology to a set with the discrete topology can be continuous. — Iosif Pinelis, Feb 24 '23 at 18:23
I do not have enough intuition to think of correspondences as mappings from sets to collections of sets so I am not sure what an appropriate topology would be. — Ded, Feb 24 '23 at 18:26
Upper and lower hemicontinuity make sense without a topology on sets. That being said, the Vietoris topology does the job. — Michael Greinecker, Feb 24 '23 at 18:37
@Ded As Michael says, given a correspondence (=relation) $R\colon X⇸Y$ you can put the upper or lower Vietoris topology in $\mathcal{P}(Y)$ and then define continuitiy/openness/closedness of $R$ via its adjunct $R^\dagger\colon X\to\mathcal{P}(Y)$. A reference for openness/closedness in this context is Clementino–Tholen, p. 9, while one for upper/lower semicontinuity for total relations is Klein–Thompson's Theory of Correspondences, Theorems 7.1.4 and 7.1.7. — Emily, Feb 25 '23 at 08:12

score 2 · Accepted Answer · answered Feb 24 '23 at 18:55

Neither implication holds.

Let $\Theta=\mathcal{X}=[-1,1]$.

First, define $f$ by $f(x,y)=xy$. Then $C$ is not lower hemicontinuous. Indeed, $$\big\{\theta\mid C(\theta)\cap (-1,0)\neq\emptyset\big\}=[-1,0]$$ is not open.

Next, let $f$ be any discontinuous function with nonnegative values. Then $C$ is constant with value $[-1,1]$ and, therefore, continuous.

Can continuous correspondence be represented via continuous functions?

1 Answers1