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 =.

osc.png

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 =.

isc.png

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}.

continuous

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

cvx-valued-isc

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:

inverse-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.

osc-no-continuous-selection

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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One response

  1. […] Under these assumptions, the mapping is outer semi-continuous. […]

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