# Local boundedness of set-valued functions

Local boundedness is an important property of set-valued functions and it is the missing piece of the puzzle in the study of semicontinuity. A function is locally bounded at a point $x$ if the images of some neighbourhoods of this point are bounded sets. We see how the assumption of local boundedness leads to some very useful properties related to semicontinuity of $F$, of its inverse $F^{-1}$ and more.

We start by stating the definition of local boundedness:

Definition 1 (Local boundedness). Let $F:\mathbb{R}^n \rightrightarrows \mathbb{R}^m$ be a set-valued mapping. We say that it is locally bounded at $x$ if there is a neighbourhood of $x$, $V_v$ so that the set $F(V)$ is bounded. We say that $F$ is locally bounded in $X\subseteq \mathbb{R}^n$ if it is locally bounded at every $x\in X$. We say that it is (uniformly) bounded if its range is a bounded set.

First, note that if $V_v$ so that the set $F(V)$ is bounded, the same holds for any other neighbourhood $U_x$ of $x$ with $U_x\subseteq V_x$. Second, it follows from the definition that if $F$ is locally bounded at $x$ it is bounded-valued at $x$. The opposite is not correct; take for example the single-valued function $F(x) = x^{-1}$ for $x\neq 0$ and $F(0) = 0$. This function is not locally bounded at the origin. We give a slightly more intricate example of a set-valued function which is nowhere locally bounded in Fugure 1 using an enumeration of the rationals $\mathbb{Q} = \{q_n\}_n$.

Figure 1. A set-valued function which is bounded-valued but not locally bounded. This function is, in fact, continuous.

The definition of local boundedness can be also stated using open balls instead of abstract open sets: $F:\mathbb{R}^n \rightrightarrows \mathbb{R}^m$ is locally bounded at $x$ if there is a $\mathcal{B}(x,\rho)$ – for some $\rho > 0$ so that $F(\mathcal{B}(x,\rho))$ is bounded. Equivalently, it is locally bounded at $x$ if there are $\rho, \delta>0$ so that $\|y\|<\rho$ whenever $y\in F(x’)$ for all $x'$ with $\|x'-x\|< \delta$.

It is easy to see that the locally bounded functions are exactly the ones which map bounded sets to bounded sets (preserve boundedness). It is then possible to derive a simple criterion for local boundedness using sequences: $F:\mathbb{R}^n \rightrightarrows \mathbb{R}^m$ is locally bounded if $\{u_n\}_n$ is bounded where $u_n\in F(x_n)$ whenever $\{x_n\}_n$ is a bounded sequence.

The inverse of a locally bounded function with bounded domain is definitely locally bounded. In general, we have the following proposition

Proposition 2 (Local boundedness of inverses). Let $F:\mathbb{R}^n \rightrightarrows \mathbb{R}^m$ be a locally bounded function. Then $F^{-1}$ is locally bounded if and only if

$\|x_n\|\to \infty, u_n\in F(x_n) \Rightarrow \|u_n\|\to \infty$

Local boundedness is a useful condition which avails us the following outer semicontinuity conditions

Proposition 3 (Conditions for OSC under local boundedness). Suppose $F:\mathbb{R}^n \rightrightarrows \mathbb{R}^m$ is locally bounded at $x_0$. Then $F$ is osc at $x_0$ if and only if $F(x_0)$ is closed and for every open set $O \supset F(x_0)$, there is a neighbourhood $V$ of $x_0$ such that $F(V) \subseteq O$.

Proposition 3 is reminiscent of a similar result regarding continuity of signle-valued functions in topological spaces where $O\supset F(x_0)$ may be interpreted as a neighbourhood of $F(x_0)$ (in a topology of the powerset of $\mathbb{R}^m$).

Bibliographic notes

Proofs and further details related to this post can be found in [1] and some basic facts are given in [3]. Some very interesting notes on the property of local boundedness for single-valued functions are presented in [2].

[1] R.T. Rockafellar and R.J-B. Wets, Variational Analysis, Grundlehren der mathematischen Wissenshaften, vol. 317, Springer, Dordrecht 2000, ISBN: 978-3-540-62772-2.
[2] R.A. Mimma, Locally bounded functions, Real Analysis Exchange 23(1), 1998-99, pp. 251-158, Available online at http://projecteuclid.org/download/pdf_1/euclid.rae/1337086093
[3] Locally bounded: http://planetmath.org/locallybounded, at planetmat.org

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