Inner and Outer Semicontinuity of Multifunctions

Where in this post we introduce the notions of inner and outer semicontinuity for multi-valued functions using the Painlevé-Kuratowski notion of set-convergence which we introduced previously. We then provide a few useful characterisations of (semi)continuity for multi-valued mappings. We discuss the similarities and differences between these definitions and there counterparts for single-valued functions.

Let F:X\rightrightarrows Y, where X, Y are Hausdorff topological spaces (or metric spaces), be a set-valued function, i.e., to every x\in X, it assigns a set F(x)\subseteq Y which can be the empty set as well. We define the domain of F to be the set \mathrm{dom} F = \{x\in X: F(x) \neq \varnothing\}.

Set-valued functions define inner and outer limits as x \to x_0; this is how:

\limsup_{x\to x_0} F(x) = \bigcup_{x^\nu \to x_0}\limsup_{\nu\to\infty} F(x^\nu),

where, F(x^\nu) defines a sequence of sets and \limsup_{\nu\to\infty} F(x^\nu) is the outer limit of this sequence; the union in the above formula is taken with respect to all converging sequences x^\nu \to x.

What is the same, we define the inner limit of F:

\liminf_{x\to x_0} F(x) = \bigcap_{x^\nu \to x_0}\liminf_{\nu\to\infty} F(x^\nu),

Let us give the definitions of inner and outer semicontinuity for set-valued mappings:

Definition 1 (Outer Semicontinuity). A set-valued mapping F:X\rightrightarrows Y is outer semicontinuous (OSC) at x_0 if

\limsup_{x\to x_0}F(x) \subseteq F(x_0)

where we may well replace \subseteq with =.


Figure 1. Outer semicontinuous function at x.


The reason why the inclusion can be replaced by a simple equality in the definition of OSC functions is that the constant sequence x^\nu = x has limit \mathrm{cl} F(x) – Indeed,

F(x) \subseteq \mathrm{cl} F(x) \subseteq \limsup_{\nu, x^\nu\to x_0} F(x^\nu)\subseteq \limsup_{x\to x_0}F(x).

Similary, we give the definition of inner semicontinuous set-valued functions:

Definition 2 (Inner semicontinuity). A set-valued mapping F:X\rightrightarrows Y is inner semicontinuous (OSC) at x_0 if

\liminf_{x\to x_0}F(x) \supseteq F(x_0)

where if F is closed-valued we may well replace \subseteq with =.


Figure 2. Illustration of an ISC set-valued function F at a point x. Notice that the value F(x) is shown on the y-axis for clarity.

It is easy to see from the definition that S:X\rightrightarrows Y is osc everywhere if any only if its graph, i.e., the set \mathrm{gph} S = \{(x,y): y\in S(x)\} is a closed set in X\times Y.

Definition 3 (Continuity). A set-valued mapping F:X\rightrightarrows Y is continuous at \bar{x} if it is both inner and outer semicontinuous at \bar{x}.


Figure 3. The graph of a function which is continuous at x2, but not continuous (neither osc nor isc) at x1.

According to Definition 3, a function F:X\rightrightarrows Y is continuous at at \bar{x} if \lim_{x\to \bar{x}} exists and is equal to F(\bar{x}). This implies that F(\bar{x}) is a closed set.

For a (proper) function f:\mathbb{R}^n \to \mathbb{R}, its epigraphical profile is defined as

E_f (x) = \{\alpha\in \bar{\mathbb{R}}: f(x) \leq \alpha\}

is a set-valued function whose graph is the epigraph of f. Clearly, E_f is osc everywhere if and only if f is lsc (lower semi-continuous everywhere); recall, that a function is lsc if and only if its graph is closed.

For closed-valued functions, outer semicontinuity becomes the property of inverting compact sets to closed sets. We state this in the following proposition:

Proposition (OSC for closed-valued functions). Let F: X\rightrightarrows Y be a closed-valued function. It is osc if and only if for every compact set B, F^{-1}(B) is closed.

A key property which allows us to prove Proposition 1 is the fact that y\in F(x) if and only if x\in F^{-1}(y). This allows us to prove that a function is osc if and only if its inverse if osc.

Inner semicontinuity can be established by checking whether a function inverts all open sets to open sets. This is a bit reminiscent of the topological definition of continuity.

Proposition (ISC criterion). A function F: X\rightrightarrows Y is isc if and only if F^{-1}(O) is open for every open set O\subseteq Y.

Similar to the closure of an extended-real-valued function f:X\to \bar{\mathbb{R}} (which is the function whose epigraph is the closure of the epigraph of f), the closure of a set-valued function F: X\rightrightarrows Y is defined as the function \mathrm{cl} F whose graph is the closure of the graph of F.

\mathrm{gph}( \mathrm{cl} F) = \mathrm{cl} (\mathrm{gph} F)

This new set-valued function is also known the the osc-hull of F and

\mathrm{gph}( \mathrm{cl} F)(x) = \limsup_{s\to x}F(s)

Basically, we may construct osc mappings from any set-valued mapping. It is, however, not straightforward to construct isc mappings in a similar way. Contrary to what we would expect, the mapping defined by

\tilde{F}(x) = \liminf_{s\to x}F(s)

may fail to be isc. Inner semicontinuity is usually difficult to establish. However, in the case when the function is convex-valued, Rockafellar and Wets provide the following very useful theorem:

Theorem 1 (ISC for convex-valued functions). Let F:X \rightrightarrows Y be a set-valued mapping with domain \mathrm{dom}F = \{x\in X: F(x) \neq \varnothing\} and let \bar{x}\in X. Then,

(i) [see also Fig. 4] Suppose that F is convex-valued and \mathrm{int} S(\bar{x})\neq \varnothing. Then, F is isc with respect to \mathrm{dom}F if and only if for all u\in \mathrm{int} F(\bar{x}) there is a neighbourhood W of the point (\bar{x},u) such that W\cap (\mathrm{dom}F\times Y)\subseteq \mathrm{gph}F


Figure 4. A convex-valued function which is isc at at point x-bar.

(ii) The function is isc at \bar{x} if and only if (\bar{x}, u)\in \mathrm{int}(\mathrm{gph} F) for all u \in \mathrm{int} F(\bar{x})

(iii) If F is graph-convex, i.e., the graph of F is a convex set, and \bar{x}\in\mathrm{int}(\mathrm{dom}F), then F is isc at this point.

A notable feature of inner semicontinuity is that the inverse of an isc function may well fail to be isc – even when the function is closed-valued and/or convex-valued. We know that for closed-valued functions, the outer semicontinuity of F entails the outer semicontinuity of F^{-1}. In Figure 5 we give an example of an isc function whose inverse is not isc:


Figure 5. An inner semicontinuous function whose inverse is not inner semicontinuous.

There is a connection between lower/upper semicontinuity for single-valued functions and inner/outer semicontinuity for set-valued functions, but one should be wary: for example, a single-valued function F:X\to Y is continuous at x if and only if it is isc at x. If F is extended-real-valued then the outer semicontinuity of its epigraphical profile is equivalent to F being lsc.

It is quite interesting to note that closed-valued inner semicontinuous mappings with a σ-compact domain admit continuous selections , that is if S:\mathbb{R}^n\rightrightarrows \mathbb{R}^m is isc, then there is a continuous single-valued function s:\mathbb{R}^n \to \mathbb{R}^m with s(x) \in S(x) for every x\in \mathrm{dom} S. Outer semicontinuous mappings do not necessarily admit a continuous selection as we may see in Figure 6.


Figure 6. OSC (blue) mappings do not necessarily have continuous selections. Michael’s theorem, instead, provides conditions for ISC (green) mappings to have continuous selections.


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