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 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 , of its inverse and more.
We start by stating the definition of local boundedness:
Definition 1 (Local boundedness). Let be a set-valued mapping. We say that it is locally bounded at if there is a neighbourhood of , so that the set is bounded. We say that is locally bounded in if it is locally bounded at every . We say that it is (uniformly) bounded if its range is a bounded set.
First, note that if so that the set is bounded, the same holds for any other neighbourhood of with . Second, it follows from the definition that if is locally bounded at it is bounded-valued at . The opposite is not correct; take for example the single-valued function for and . 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 .
The definition of local boundedness can be also stated using open balls instead of abstract open sets: is locally bounded at if there is a – for some so that is bounded. Equivalently, it is locally bounded at if there are so that whenever $y\in F(x’)$ for all with .
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: is locally bounded if is bounded where whenever 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 be a locally bounded function. Then is locally bounded if and only if
Local boundedness is a useful condition which avails us the following outer semicontinuity conditions
Proposition 3 (Conditions for OSC under local boundedness). Suppose is locally bounded at . Then is osc at if and only if is closed and for every open set , there is a neighbourhood of such that .
Proposition 3 is reminiscent of a similar result regarding continuity of signle-valued functions in topological spaces where may be interpreted as a neighbourhood of (in a topology of the powerset of ).
Proofs and further details related to this post can be found in  and some basic facts are given in . Some very interesting notes on the property of local boundedness for single-valued functions are presented in .
 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.
 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
 Locally bounded: http://planetmath.org/locallybounded, at planetmat.org