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 V is a finite dimensional real normed linear space equipped with a norm :VR. It is also equipped with a metric d(x,y)=xy. Wherever necessary, it is also equppied with an real inner product ,:V×VR.

We recall that a function f:VR is L-Lipschitz if

|f(x)f(y)|Lxyx,yV.

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 f:VR with S=domf be a function.

Let a be an accumulation point of S. The limit superior of f at a is defined by

lim supxaf(x)=infδ>0supxBd(a,r)Sf(x).

The limit inferior of f at a is defined by

lim infxaf(x)=supδ>0infxBd(a,r)Sf(x).

f is lower-semicontinuous at aS if for every ϵ>0, there exists δ>0 such that

f(a)ϵ<f(x) for every xB(a,δ)S.

f is upper-semicontinuous at aS if for every ϵ>0, there exists δ>0 such that

f(x)<f(a)+ϵ for every xB(a,δ)S.

We say that f is lower semicontinuous (l.s.c.) if f is l.s.c. at every point of S. Similarly, we say that f is upper semicontinuous (u.s.c.) if f is u.s.c. at every point of S.

f is continuous at aS if and only if f is l.s.c. as well as u.s.c. at a.

Let aS be an accumulation point of S. Then, f is lower semicontinuous at a if and only if

lim infxaf(x)f(a).

Similarly, f is upper semicontinuous at a if and only if

lim supxaf(x)f(a).

Let aS. Then, f is l.s.c. at a if and only if for every sequence {xn} of S that converges to a,

lim infnf(xn)f(a).

Similarly, f is upper semicontinuous at a if and only if every sequence {xk} of S that converges to a,

lim supnf(xn)f(a).

The following conditions are equivalent.

  1. f is l.s.c.

  2. f is closed; i.e., every sublevel set of f is closed with respect to the subspace topology (S,).

  3. epif is closed in VR.

9.13.1. Closure#

Definition 9.63 (Closure of a convex function)

Let f:VR be a convex function. Then, its closure is defined to be the lower semicontinuous hull of f.

If g is the closure of f, then

epig=clepif.

If f:V(,] is a proper convex function, then also, its closure is defined to be the lower semicontinuous hull.

If f:VR is an improper convex function which attains a value f(x)= at some xV, then its closure is defined to be the constant function g(x)= for all xV.

The closure of a convex function is denoted by clf.

Theorem 9.172 (Closure of a convex function is convex)

The closure of a convex function is convex.

Proof. Let f be a convex function. Then, epif is convex. Let g be its closure.

  1. If f is improper, then g is a constant function, hence convex.

  2. Otherwise, epig=clepif.

  3. Since f is convex, hence epif is convex.

  4. Due to Theorem 9.123, the closure of a convex set is convex.

  5. Hence, epig is convex.

  6. Hence, g is convex.

Definition 9.64 (Closed convex function)

A convex function f:VR is called closed if

clf=f.

For a proper convex function, closedness is same as lower semicontinuity.

The only closed improper convex functions are

  1. f(x)=xV. Here epif=.

  2. f(x)=xV. Here epif=V×R.

Example 9.62 (Closed convex function with open domain)

Let f:R(,] be given as

f(x)={1x,x>0,x0.

Then, domf=(0,).

  1. domf is an open interval in R.

  2. f is continuous at every x>0.

  3. Thus, f is l.s.c. at every x>0.

  4. Thus, f is l.s.c.

  5. Let the sublevel set for rR be given by

    Tr={x(0,)|f(x)r}.
  6. We can see that Tr= for r0.

  7. For r>0,

    f(x)=1xrx1r.
  8. Thus, Tr=[1r,).

  9. Tr is indeed closed in the topology (domf,||).

  10. Since f is l.s.c., hence epif is closed.

  11. Thus, clf=f.

  12. Thus, f is a closed convex function.

Remark 9.11 (Closure of proper convex functions and epigraph)

Let f:V(,] be a proper convex function. Then,

epiclf=clepif.

This follows from the definition, since g=clf is defined by the fact that

epig=clepif.

9.13.2. Continuity#

Recall from Definition 3.38 that a function f:V(,] is continuous at a point adomf if for every ϵ>0, there exists δ>0 (depending on ϵ and a) such that for every xdomf

xa<δ|f(x)f(a)|<ϵ

holds true. In other words,

|f(x)f(a)|<ϵ for every xB(a,δ)S.

Convex functions are not necessarily continuous on non-open sets.

Example 9.63 (A convex function which is not continuous)

Let f:RR be given by

f(x)={1,x=0,x20<x1.

We can see that domf=[0,1]. f is continuous on (0,1) but f is not continuous (from the right) at x=0. It is continuous (from the left) at x=1.

Convex functions are continuous at points in the interior of their domain.

9.13.2.1. Continuity of Univariate Closed Convex Functions#

Theorem 9.173 (Continuity of closed convex univariate functions)

Let f:R(,] be a proper closed and convex function. Then, f is continuous over domf.

Proof. Since f is convex, hence its domain is convex. Hence domf must be an interval I.

  1. If intI=, then I must be a singleton.

  2. In that case f is continuous obviously.

  3. Now consider the case where intI.

  4. Then, due to Theorem 9.174, f is continuous at every xintI.

  5. If I is open (i.e., it has no endpoints), then there is nothing more to prove.

  6. We are left with showing the (one sided) continuity of f at one of the endpoints of I if it has any.

  7. Since, the argument will be identical for either of the endpoints, without loss of generality, let us assume that I has a left endpoint a and we show the continuity from the right at a; i.e. limxa+f(x)=f(a).

  8. Pick any cI such that c>a.

  9. Define a function

    g(t)=f(ct)f(c)t.
  10. Clearly, g is defined over (0,ca].

  11. We shall show that g is nondecreasing and upper bounded over (0,ca].

  12. Pick any t,s satisfying 0<tsca.

  13. Then,

    ct=(1ts)c+ts(cs).
  14. ts is well defined and ts(0,1].

  15. Thus, ct is a convex combination of c and cs.

  16. Since f is convex, hence

    f(ct)(1ts)f(c)+tsf(cs)f(ct)f(c)ts(f(cs)f(c))f(ct)f(c)tf(cs)f(c)s.
  17. Thus,

    g(t)g(s)0<tsca.
  18. Thus, g is nondecreasing over (0,ca].

  19. Finally g(ca)=f(a)f(c)ca is finite since both c,adomf.

  20. Since g is nondecreasing, hence

    g(t)g(ca)t(0,ca].
  21. Thus, g is upper bounded.

  22. Since g is nondecreasing and upper bounded, hence due to Theorem 2.38, the left hand limit of g(t) at ca exists and is equal to some real number, say,

    limt(ca)g(t)=.

    Note that we haven’t said that g is continuous from the left at ca.

  23. Recall from the definition of g that

    f(ct)=f(c)+tg(t).
  24. Hence

    limt(ca)f(ct)=f(c)+(ca).
  25. Replacing ct by r, we get

    limra+f(r)=f(c)+(ca).
  26. We have shown so far that the limit from the right at a exists for f and is equal to f(c)+(ca).

  27. Using the upper bound on g, we can say that

    f(ct)=f(c)+tg(t)f(c)+(ca)g(ca)=f(c)+f(a)f(c)=f(a)

    holds true for every t(0,ca].

  28. Thus,

    limra+f(r)=limt(ca)f(ct)f(a).
  29. On the other hand, since f is closed, hence it is also lower semicontinuous. This means that

    limra+f(r)f(a).
  30. Combining these two inequalities, we get

    limra+f(r)=f(a).
  31. Thus, f is indeed continuous from the right at a.

  32. Similarly, if I has a right endpoint b, then f is continuous from the left at b.

  33. Thus, f is continuous at every point in its domain.

9.13.2.2. Local Lipschitz Continuity#

Theorem 9.174 (Local Lipschitz continuity of convex functions)

Let V be an n-dimensional real normed linear space. Let f:VR be a convex function with S=domf. Let aintS. Then, there exists r>0 and L>0 such that B(a,r)S and

|f(x)f(a)|Lxa

for every xB[a,r].

In other words, f is locally Lipschitz continuous at every interior point of its domain.

We recall from Theorem 9.125 that if dimaffS<n then S has an empty interior. Thus, if intS is nonempty, then, affS=V.

Proof. We shall structure the proof as follows. For any aintS:

  1. We show that f is bounded on a closed ball B[a,r]S.

  2. Then, we show that f satisfies the Lipschitz inequality |f(x)f(a)|Lxa on the closed ball B[a,r] for a specific choice of L depending on a and r.

We first introduce norm on V and describe its implications.

  1. Choose a basis B={e1,,en} for V.

  2. For every xV, we have a unique representation

    x=i=1nxiei.
  3. Let T:VRn be a coordinate mapping which maps every vector xV to its coordinate vector (x1,,xn)Rn.

  4. T is an isomorphism.

  5. Define :VR as

    x=T(x)=maxi=1,,n|xi|.
  6. It is easy to show that is a norm on V.

  7. By Theorem 4.60, all norms are equivalent.

  8. Thus, and are equivalent norms for V.

  9. By Definition 4.69, the norm topology is identical for all norms in a finite dimensional space.

  10. Thus, a point is an interior point of S irrespective of the norm chosen.

  11. We introduce the closed and open balls in (V,) as

    B[a,δ]={xV|xaδ} and B(a,δ)={xV|xa<δ}.
  12. Let aintS.

  13. Then, there exists r1>0 such that B[a,r1]S. due to Remark 3.2.

  14. Then, B(a,r1)B[a,r1].

  15. By Theorem 3.25, there is an r2>0 such that B(a,r2)B(a,r1).

  16. By Theorem 3.2, we can pick an 0<r<r2 such that

    B[a,r]B(a,r2)B(a,r1)B[a,r1]S.

We now show that f is bounded on B[a,r].

  1. B[a,r1] is closed and bounded.

  2. Hence B[a,r1] is compact due to Theorem 4.66.

  3. By Krein Milman theorem, a compact convex set is convex hull of its extreme points. Thus,

    B[a,r1]=convextB[a,r1].
  4. Let v1,,vN be the N=2n extreme points of B[a,r1].

    1. These extreme points are given by

      vi=a+r1wi

      where wi are the vectors with coordinates {1,1}n.

    2. In other words,

      wi=j=1nwjiej

      where wji{1,1}.

    3. Note that

      via=rwi=r1max{|wji|}=r1.
    4. Thus, vibdB[a,r1].

    5. Readers can verify that these are indeed the extreme points of B[a,r1] and there are no other extreme points.

  5. Then, by Krein Milman theorem,

    B[a,r1]=conv{v1,,vN}.
  6. Thus, every xB[a,r1] is a convex combination of the extreme points. Specially, there exists tΔN (unit simplex of RN) such that

    x=i=1Ntivi.
  7. Now, by Jensen's inequality,

    f(x)i=1Ntif(vi).
  8. Let M=max{f(v1),,f(vN)}.

  9. Then,

    f(x)i=1Ntif(vi)i=1NtiM=Mi=1Nti=M.
  10. Since B[a,r]B[a,r1], hence f(x)M for every xB[a,r].

We have shown that f(x)M for every xB[a,r]. We next find an L such that

f(x)f(a)Lxa

for every xB[a,r].

  1. Let xBd[a,r] (the deleted neighborhood).

  2. Then, xar and f(x)M.

  3. Let α=1rxa. Note that by definition α1.

  4. Define

    y=a+1α(xa).
  5. Note that

    ya=rxaxa=r.
  6. Thus, yB[a,r].

  7. Hence f(y)M.

  8. We can rewrite the above equation (defining y) as

    x=αy+(1α)a.
  9. Thus, x is a convex combination of y,a.

  10. Then, by convexity,

    f(x)αf(y)+(1α)f(a)=f(a)+α(f(x)f(a))f(a)+α(Mf(a))=f(a)+1rxa(Mf(a)).
  11. Consequently,

    f(x)f(a)Mf(a)rxa.
  12. Let L=Mf(a)r.

  13. Then, for every xB[a,r], we have

    f(x)f(a)Lxa.

We next show that for this choice of L

f(a)f(x)Lxa

for every xB[a,r].

  1. Define

    z=a+1α(ax).
  2. It is easy to see that az=r.

  3. Hence, zB[a,r] and f(z)M.

  4. Rearranging, we have

    x=a+α(az).
  5. Now, note that:

    a=11+α(a+α(az))+α1+αz.
  6. Thus, a is a convex combination of x=a+α(az) and z.

  7. Also, both x,zB[a,r].

  8. Applying convexity,

    f(a)11+αf(x)+α1+αf(z).
  9. Thus,

    (1+α)f(a)f(x)+αf(z).
  10. Thus,

    f(x)(1+α)f(a)αf(z)=f(a)+α(f(a)f(z)).
  11. Continuing from here

    f(x)f(a)+α(f(a)f(z))f(a)α(Mf(a))=f(a)Mf(a)rxa=f(a)Lxa.
  12. Thus, f(a)f(x)Lxa.

Combining, we see that with L=Mf(a)r,

|f(x)f(a)|Lxa

for every xB[a,r].

Thus, f is locally Lipschitz at every interior point of S=domf.

9.13.2.3. Continuity of Proper Convex Functions#

Theorem 9.175 (Continuity of proper convex functions)

Let V be an n-dim real normed linear space. Let f:VR be a real valued convex function. Then it is continuous. More generally, if f:V(,] is a proper convex function, then f, restricted to domf, is continuous over the relative interior of domf.

Proof. We will restrict our attention to the affine hull of the domain of f. We shall assume that origin belongs to ridomf. We shall use a transformation argument as needed.

  1. Let S=domf.

  2. Let A=affS.

  3. Let L the linear subspace parallel to A.

  4. Let ariS. If 0riS, we can pick a=0.

  5. Define the function

    g(x)=f(x+a).
  6. Clearly g(0)=f(a).

  7. We can see that domg=Sa and affdomg=L.

  8. Consider the restriction of g to L defined as h:L(,] given by h(x)=g(x).

  9. domh=Sa.

  10. We note that relative interior of Sa w.r.t. V is the same as the interior of Sa w.r.t. L.

  11. By Theorem 9.174, h is locally Lipschitz continuous at every interior point of Sa (relative to L).

  12. Hence g is continuous over the relative interior of Sa.

  13. Hence f is continuous over the relative interior of S.

Corollary 9.12 (Closedness of real valued convex functions)

Let V be an n-dim real normed linear space. A real valued convex function f:VR with V=domf is closed.

Proof. By Theorem 9.175, f is continuous over V.

  1. For real valued convex functions, V=domf.

  2. V is a closed set in the topology of (V,).

  3. By Theorem 3.96, continuity and closed domain imply that f is closed.