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.

 

bd-valued-not-locbdd

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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