Continuity
Contents
9.13. Continuity#
This section focuses on topological properties of convex functions in normed linear spaces. In particular, we discuss closure of convex functions, continuity of convex functions at interior points.
Main references for this section are [6, 17, 67].
Throughout this section, we assume that
We recall that a function
The concept of semicontinuity, inferior and superior limits and closedness of real valued functions in metric spaces is discussed in detail in Real Valued Functions.
We recall some results.
Let
Let
The limit inferior of
We say that
Let
Similarly,
Let
Similarly,
The following conditions are equivalent.
is l.s.c. is closed; i.e., every sublevel set of is closed with respect to the subspace topology . is closed in .
9.13.1. Closure#
(Closure of a convex function)
Let
If
If
If
The closure of a convex function is denoted by
(Closure of a convex function is convex)
The closure of a convex function is convex.
Proof. Let
If
is improper, then is a constant function, hence convex.Otherwise,
.Since
is convex, hence is convex.Due to Theorem 9.123, the closure of a convex set is convex.
Hence,
is convex.Hence,
is convex.
(Closed convex function)
A convex function
For a proper convex function, closedness is same as lower semicontinuity.
The only closed improper convex functions are
. Here . . Here .
(Closed convex function with open domain)
Let
Then,
is an open interval in . is continuous at every .Thus,
is l.s.c. at every .Thus,
is l.s.c.Let the sublevel set for
be given byWe can see that
for .For
,Thus,
. is indeed closed in the topology .Since
is l.s.c., hence is closed.Thus,
.Thus,
is a closed convex function.
(Closure of proper convex functions and epigraph)
Let
This follows from the definition, since
9.13.2. Continuity#
Recall from Definition 3.38 that
a function
holds true. In other words,
Convex functions are not necessarily continuous on non-open sets.
(A convex function which is not continuous)
Let
We can see that
Convex functions are continuous at points in the interior of their domain.
9.13.2.1. Continuity of Univariate Closed Convex Functions#
(Continuity of closed convex univariate functions)
Let
Proof. Since
If
, then must be a singleton.In that case
is continuous obviously.Now consider the case where
.Then, due to Theorem 9.174,
is continuous at every .If
is open (i.e., it has no endpoints), then there is nothing more to prove.We are left with showing the (one sided) continuity of
at one of the endpoints of if it has any.Since, the argument will be identical for either of the endpoints, without loss of generality, let us assume that
has a left endpoint and we show the continuity from the right at ; i.e. .Pick any
such that .Define a function
Clearly,
is defined over .We shall show that
is nondecreasing and upper bounded over .Pick any
satisfying .Then,
is well defined and .Thus,
is a convex combination of and .Since
is convex, henceThus,
Thus,
is nondecreasing over .Finally
is finite since both .Since
is nondecreasing, henceThus,
is upper bounded.Since
is nondecreasing and upper bounded, hence due to Theorem 2.38, the left hand limit of at exists and is equal to some real number, say,Note that we haven’t said that
is continuous from the left at .Recall from the definition of
thatHence
Replacing
by , we getWe have shown so far that the limit from the right at
exists for and is equal to .Using the upper bound on
, we can say thatholds true for every
.Thus,
On the other hand, since
is closed, hence it is also lower semicontinuous. This means thatCombining these two inequalities, we get
Thus,
is indeed continuous from the right at .Similarly, if
has a right endpoint , then is continuous from the left at .Thus,
is continuous at every point in its domain.
9.13.2.2. Local Lipschitz Continuity#
(Local Lipschitz continuity of convex functions)
Let
for every
In other words,
We recall from Theorem 9.125
that if
Proof. We shall structure the proof as follows. For any
We show that
is bounded on a closed ball .Then, we show that
satisfies the Lipschitz inequality on the closed ball for a specific choice of depending on and .
We first introduce
Choose a basis
for .For every
, we have a unique representationLet
be a coordinate mapping which maps every vector to its coordinate vector . is an isomorphism.Define
asIt is easy to show that
is a norm on .By Theorem 4.60, all norms are equivalent.
Thus,
and are equivalent norms for .By Definition 4.69, the norm topology is identical for all norms in a finite dimensional space.
Thus, a point is an interior point of
irrespective of the norm chosen.We introduce the closed and open balls in
asLet
.Then, there exists
such that . due to Remark 3.2.Then,
.By Theorem 3.25, there is an
such that .By Theorem 3.2, we can pick an
such that
We now show that
is closed and bounded.Hence
is compact due to Theorem 4.66.By Krein Milman theorem, a compact convex set is convex hull of its extreme points. Thus,
Let
be the extreme points of .These extreme points are given by
where
are the vectors with coordinates .In other words,
where
.Note that
Thus,
.Readers can verify that these are indeed the extreme points of
and there are no other extreme points.
Then, by Krein Milman theorem,
Thus, every
is a convex combination of the extreme points. Specially, there exists (unit simplex of ) such thatNow, by Jensen's inequality,
Let
.Then,
Since
, hence for every .
We have shown that
for every
Let
(the deleted neighborhood).Then,
and .Let
. Note that by definition .Define
Note that
Thus,
.Hence
.We can rewrite the above equation (defining
) asThus,
is a convex combination of .Then, by convexity,
Consequently,
Let
.Then, for every
, we have
We next show that for this choice of
for every
Define
It is easy to see that
.Hence,
and .Rearranging, we have
Now, note that:
Thus,
is a convex combination of and .Also, both
.Applying convexity,
Thus,
Thus,
Continuing from here
Thus,
.
Combining, we see that with
for every
Thus,
9.13.2.3. Continuity of Proper Convex Functions#
(Continuity of proper convex functions)
Let
Proof. We will restrict our attention to the affine hull
of the domain of
Let
.Let
.Let
the linear subspace parallel to .Let
. If , we can pick .Define the function
Clearly
.We can see that
and .Consider the restriction of
to defined as given by . .We note that relative interior of
w.r.t. is the same as the interior of w.r.t. .By Theorem 9.174,
is locally Lipschitz continuous at every interior point of (relative to ).Hence
is continuous over the relative interior of .Hence
is continuous over the relative interior of .
(Closedness of real valued convex functions)
Let
Proof. By Theorem 9.175,
For real valued convex functions,
. is a closed set in the topology of .By Theorem 3.96, continuity and closed domain imply that
is closed.