JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi 10.1155/2019/9602504 9602504 Research Article Distance to Spaces of Semicontinuous and Continuous Functions http://orcid.org/0000-0001-7592-6121 Angosto Carlos 1 Martin Miguel Departamento de Matemática Aplicada y Estadística Universidad Politécnica de Cartagena Spain upct.es 2019 852019 2019 22 03 2019 16 04 2019 852019 2019 Copyright © 2019 Carlos Angosto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Given a topological space X, we establish formulas to compute the distance from a function fRX to the spaces of upper semicontinuous functions and lower semicontinuous functions. For this, we introduce an index of upper semioscillation and lower semioscillation. We also establish new formulas about distances to some subspaces of continuous functions that generalize some classical results.

Ministerio de Economía y Competitividad MTM2017-83262-C2-2-P Fundación Séneca 20906/PI/18
1. Introduction

Several classical and modern results deal with distances, optimization, equalities, and inequalities. For this, several authors have studied how to compute distances to some spaces of functions, for example, spaces of continuous functions (), spaces of Baire-one functions ([2, 3]), and spaces of measurable functions and integrable functions (). This kind of results has been used in a big number of papers (for instance, ).

Our aim here is to establish analogous formulas to study distances to spaces of upper semicontinuous functions and lower semicontinuous functions and to study distances to some subspaces of continuous functions that generalize the mentioned result in .

Recall that if X is a topological space, a function fRX is upper (resp., lower) semicontinuous if for all xX and ε>0 there is U, a neighborhood of x, such that f(z)<f(x)+ε (resp., f(z)>f(x)-ε) for all zU. Observe that f is upper (resp., lower) semicontinuous if and only if f-1([t,+)) (resp., f-1((-,t])) is a closed set for all tR.

Throughout the paper X is a topological space, C(X) denotes the subspace of RX made up of all continuous functions, usC(X) denotes the subspace of upper semicontinuous functions, and lsC(X) denotes the subspace of lower semicontinuous functions. For fRX and CRX we denote (1)df,C=infgCdf,gwhere (2)df,g=supxXfx-gxis the supremum metric (that can take the value +).

In Section 2 we introduce the indexes of upper and lower semioscillation and use them to study the distances of a function to the spaces of upper semicontinuous functions and lower semicontinuous functions; see Theorem 5.

In Section 3 we establish some relations between distances to spaces of semicontinuous functions and spaces of continuous functions.

It is known that if X is a normal space and fRX, then(3)df,CX=12oscf;see Theorem 8. In Section 4 we study the distances to some subspaces of continuous functions. Theorem 12 shows that if fRX and AX is a closed subset of the set of points of continuity of f, then we can extend f|A to a continuous function that is also a best approximation of f in C(X). Theorem 15 studies the distance from f to the continuous functions that have fixed values in some points. Both theorems generalize (3).

2. Upper and Lower Semioscillation

The oscillation of a function fRX at xX is defined as (4)oscf,x=infUUxsupdiamfUwhere Ux denotes the set of neighborhoods of x. The oscillation of a function is used in  to study distances to spaces of continuous functions (see Theorem 8). Inspired by this idea, we define the indexes of upper semioscillation and lower semioscillation.

Definition 1.

Let fRX and xX. We define the index of upper semioscillation of f in x as (5)uoscf,x=infUUxsupzUfz-fxwhere Ux denotes the set of neighborhoods of x, and the index of upper semioscillation of f as (6)uoscf=supxXuoscf,x.Analogously we define the index of lower semioscillation of f in x as (7)loscf,x=infUUxsupzUfx-fzand the index of lower semioscillation of f as (8)loscf=supxXloscf,x.

Observe that uosc(f,x)<ε (resp., losc(f,x)<ε) if and only if there exist 0<ε<ε and a neighborhood U of x such that f(z)<f(x)+ε (resp., f(z)>f(x)-ε) for all zU. In particular f is upper (resp., lower) semicontinuous if and only if uosc(f)=0 (resp., losc(f)=0).

It is very clear that the index of oscillation and the indexes of upper semioscillation and lower semioscillation are related.

Proposition 2.

If X is a topological space and fRX is a function, then for all xX we have that (9)oscf,x=uoscf,x+loscf,x,so (10)oscfuoscf+loscf.

We introduce the following known functions.

Definition 3.

For fRX we denote (11)f¯x=infUUxsupzUfz,f_x=supUUxinfzUfz.

Proposition 4.

Let fRX. Then f¯ is an upper semicontinuous function, f_ is a lower semicontinuous function, and we have that (12)f¯x=fx+uoscf,x,f_x=fx-loscf,x.

Proof.

It is known and very easy to check that f¯ is upper semicontinuous and f_ is lower semicontinuous. We also have that (13)f¯x=infUUxsupzUfz=fx+infUUxsupzUfz-fx=fx+uoscf,xand analogously f_(x)=f(x)+losc(f,x).

The following theorem is the main result of this section.

Theorem 5.

Let X be a topological space and fRX a function. Then (14)df,usCX=12uoscf,df,lsCX=12loscf.In fact there are gusC(X) and hlsC(X) such that d(f,g)=d(f,usC(X)) and d(f,h)=d(f,lsC(X)).

Proof.

We prove the theorem in the usC(X) case, and the other one can be done analogously or we can deduce it from the usC(X) case applied to -f.

Suppose that d(f,usC(X))<d. Choose gusC(X) such that d(f,g)<d. Fix xX and ε>0. Since gusC(X), there is a neighborhood U of x such that g(z)<g(x)+ε for all zU and then (15)fz-fx<gz+d-gx-d<ε+2d.Therefore, uosc(f,x)2d for all xX and all d>d(f,usC(X)) and then uosc(f)2d(f,usC(X)).

We have to prove that if uosc(f)=δ<+, there is gusC(X) such that d(f,g)δ/2. For this we use the upper semicontinuous function (16)f¯x=infUUxsupzUfz.By Proposition 4 we have that (17)fxf¯x=fx+uoscf,xfx+δ,so the upper semicontinuous function g=f¯-δ/2 satisfies that (18)fx-δ2gxfx+δ2and the proof is over.

If a sequence of upper (resp., lower) semicontinuous functions is locally uniformly convergent to a function f, then f is also upper (resp., lower) semicontinuous. Theorem 5 can be used to get a quantitative version of this result.

Proposition 6.

Let X be a topological space and (fn)n a sequence in RX locally uniformly convergent to a function f. Then (19)df,usCXliminfndfn,usCX,df,lsCXliminfndfn,lsCX.

Proof.

Fix xX and ε>0 and choose nN and U, a neighborhood of x, such that |fm(z)-f(z)|<ε for all zU and m>n. Fix m>n and choose VU, a neighborhood of x, such that (20)fmz-fmx<uoscfm,x+εfor all zV. Then (21)fz-fx=fz-fmz+fmz-fmx+fmx-fx<uoscfm,x+3εfor all zV, so uosc(f,x)uosc(fm,x)+3εuosc(fm)+3ε and then uosc(f,x)liminfuosc(fm)+3ε. Since xX and ε>0 are arbitrary, we get by Theorem 5 that d(f,usC(X))liminfnd(fn,usC(X)). Analogously d(f,lsC(X))liminfnd(fn,lsC(X)).

Example 7.

The inequalities of Proposition 6 are sharp because they become an equality when we consider constant sequences of non-semicontinuous functions (for example, fn=f=χQRR where χA is the characteristic function of A). However, in general, the equalities do not hold. Consider fn=χ{n}RR. The sequence (fn)n is locally uniformly convergent to the null function f=0 and d(fn,lsC(R))=1/2, so (22)df,lsCR=0<12=liminfndfn,lsCR.

3. Relations between Distances to Spaces of Semicontinuous and Continuous Functions

Since C(X)usC(X)lsC(X), we have that (23)df,usCXdf,CX,df,lsCXdf,CXfor all fRX. From Theorem 8 we can obtain that in some cases the distance d(f,C(X)) can be bounded using the distances d(f,usC(X)) and d(f,lsC(X)). For this we also need the following result.

Theorem 8 (see [<xref ref-type="bibr" rid="B6">1</xref>, <xref ref-type="bibr" rid="B1">16</xref>]).

Let X be a topological space. Then the following statements are equivalent:

X is normal,

for each fRX there is gC(X) such that d(f,g)=1/2osc(f),

d(f,C(X))=1/2osc(f) for each fRX.

The version of Theorem 8 that appears in  is less general. They prove that if X is a paracompact space, then the formula d(f,C(X))=1/2osc(f) holds for all bounded functions fRX and also says that this result holds for normal spaces.

Corollary 9.

Let X be a normal topological space and fRX a function. Then(24)df,CXdf,usCX+df,lsCX.If f is an upper semicontinuous function, then(25)df,CX=df,lsCX.If f is a lower semicontinuous function, then(26)df,CX=df,usCX.

Proof.

By Proposition 2osc(f)uosc(f)+losc(f) and then by Theorems 8 and 5, we have that (27)df,CX=12oscf,x12uoscf+loscf=df,usCX+df,lsCX.Observe now that if f is upper (resp., lower) semicontinuous, then osc(f)=losc(f) (resp., osc(f)=uosc(f)), so equalities (25) and (26) also follow from Theorems 8 and 5.

Remark 10.

Considering equality (26), one can think that if f is lower (resp., upper) semicontinuous, then the best approximation of f by upper (resp., lower) semicontinuous functions that appear in the proof of Theorem 5 is continuous. However, it is easy to check that it is not true. If we consider X=R and f=χ(0,+), i.e., f is the function defined by (28)fx=0if  x0,1if  x>0,then f is lower semicontinuous but (29)f¯x=infUUxsupzUfz=χ0,+=0if  x<0,1if  x0is not continuous.

4. Distances to Subspaces of Continuous Functions

In this section we study generalizations of Theorem 8. We prove that we can obtain a best approximation g of f in C(X) that preserves the value of f in some sets of continuity points and we also can force g to have fixed values at some points. First of all, we need the following known lemma that can be found in [17, Theorem 12.16].

Lemma 11.

Let X be a topological normal space. Then if g1RX is an upper semicontinuous function and g2RX is a lower semicontinuous function such that g1g2, there exists a continuous function gC(X) such that g1gg2.

Theorem 12.

Let X be a normal space, fRX, and A be a closed subset of X such that f is continuous at x for all xA. Then there is gC(X) a continuous function such that f|A=g|A and (30)df,g=12oscf=df,CX.

Proof.

Suppose that osc(f)=δ< and define h1(x)=f¯(x)-δ/2 and h2(x)=f_(x)+δ/2. By Proposition 4, h1 is upper semicontinuous and h2 is lower semicontinuous and f¯(x)-f_(x)=uosc(f,x)+losc(f,x)=osc(f,x)osc(f)=δ, so h1h2. Define (31)gix=fxif  xA,hixif  xA,for i=0,1. Clearly g1(x)g2(x). Observe that if f is continuous in x, then f¯(x)=f_(x)=f(x), so (32)h1x<g1x=fx=g2x<h2xfor  xA.Let us prove that g1 is upper semicontinuous. If xA, there is a neighborhood U of x such that g1|U=h1|U and then uosc(g1,x)=uosc(h1,x)=0. If xA, there is a neighborhood U of x such that diam(f(U))<ε (f is continuous at x) and h1(z)<h1(x)+ε for all zU. If zUA, then g1(z)=f(z)<f(x)+ε=g1(x)+ε, and if zUAc, then g1(z)=h1(z)<h1(x)+ε<g1(x)+ε, so uosc(g1)=0. Then we have that g1 is upper semicontinuous. Analogously g2 is lower semicontinuous, and since g1g2, by Lemma 11 we have that there exists a continuous function gC(X) such that (33)g1gg2.Observe that if xA, since g1(x)=f(x)=g2(x), then g(x)=f(x), so (34)fA=gA.If xA, then g1(x)=h1(x)=f¯(x)-δ/2=f(x)+uosc(f,x)-1/2osc(f), so (35)g1x-fx=uoscf,x-12oscf12oscf.Analogously |g2(x)-f(x)|1/2osc(f), so (36)fx-12oscfg1xgxg2xfx+12oscfand then (37)df,g12oscf.To finish, we have to prove that d(f,h)1/2osc(f) for all hC(X). This is true by Theorem 8, but we include the proof here to get a self-contained proof. Suppose that h is a continuous function and define d=d(f,h). For xX, choose U, a neighborhood of x, such that diam(h(U))<ε. Then (38)fz-fxfz-hz+hz-hx+hx-fx<2d+εfor all zU so osc(f)2d and then (39)df,h12oscffor all hC(X).

Remark 13.

Observe that the final part of the proof of Theorem 12 does not need the normal hypothesis, so if X is a topological space and fRX, then (40)df,CX12oscf.

Example 14.

The set A in Theorem 12 needs to be closed. Indeed consider the function f=χARR where A=(0,+). If gC(R) is a continuous function such that f|A=g|A, then g(0)=1, so (41)df,gf0-g0=1>12oscf=12.If we change the hypothesis A{xX:fiscontinuousatx} by f|A continuous, the theorem also fails. Consider now f=χ{0}RR and A={0}. If gC(R) is a continuous function such that g(0)=f(0)=1, then clearly d(f,g)1 and osc(f)=1, so d(f,g)>1/2osc(f).

Keeping the last example in mind, we can ask how far is fRX from the set of continuous functions that takes a fixed value zR in a point xX. The following theorem says that the distance that we obtain depends on M=max{|z-f¯(x)|,|z-f_(x)|}. Since f¯(x)=f(x)+uosc(f,x), f_(x)=f(x)-losc(f,x), and f_f¯, we get that we can write M as (42)M=maxz-fx+loscf,x,fx+uoscf,x-z.

Theorem 15.

Let X be a normal space, fRX, and AX be a discrete closed subset. For each xA, fix zxRX. Then there is gC(X) such that g(x)=zx for all xA and (43)df,g=max12oscf,Mwhere (44)M=supxAmaxzx-fx+loscf,x,fx+uoscf,x-zx=supxAmaxzx-f¯x,zx-f_x.

Proof.

Take δ=max{1/2osc(f),M} and define h1(x)=f¯(x)-δ and h2(x)=f_(x)+δ. By Proposition 4h1 is upper semicontinuous, h2 is lower semicontinuous, and f¯(x)-f_(x)=uosc(f,x)+losc(f,x)=osc(f,x)osc(f)2δ, so h1h2. We have that (45)h1xzxh2xfor  xA.Indeed, if zxA, then f¯(x)-zx=f(x)+uosc(f,x)-zxδ, so h1(x)=f¯(x)-δzx and analogously zxh2(x). Define (46)gix=zxif  xA,hixif  xA,for i=1,2. Let us see that g1 is upper semicontinuous. If xA, then there is U, a neighborhood of x, such that g1|U=h1|U, so uosc(g1,x)=uosc(h1,x)=0. If xA, fix ε>0 and pick U, a neighborhood of x, such that AU={x} and h1(z)<h1(x)+ε for all zU. Then (47)g1z=h1z<h1x+εzx+ε=g1x+εfor all zU{x}, so g1 is upper semicontinuous. Analogously g2 is lower semicontinuous, and since g1g2, by Lemma 11 there is a continuous function gC(X) such that (48)g1gg2.Observe that if xA, then zx=g1(x)g(x)g2(x)=zx, so g(x)=zx for all xA. Let us see that |f(x)-g1(x)|δ.

Suppose that xA. Then (49)fx-g1x=fx-zxmaxzx-fx+loscf,x,fx+uoscf,x-zxδ.

Suppose that xA and f(x)g1(x). Then (50)g1x-fx=f¯x-δ-fx=fx+uoscf,x-δ-fx=uoscf,x-δuoscf,x-12oscf12oscfδ.

Suppose that xA and f(x)g1(x). Then (51)fx-g1xf¯x-g1x=f¯x-f¯x-δ=δ.

Thus, |f(x)-g1(x)|δ and analogously |f(x)-g2(x)|δ. We have that (52)fx-δg1xgxg2xfx+δand then (53)df,gδ.We have to prove that d(f,g)δ. Since d(f,C(X))=1/2osc(f), then d(f,g)1/2osc(f). Accordingly, the following claim finish the proof.

Claim. If h is a continuous function such that h(x)=zx for some fixed xA, then (54)df,hLmaxzx-fx+loscf,x,fx+uoscf,x-zx.Suppose without loss of generality that L=zx-f(x)+losc(f,x). Fix ε>0; then there is U, a neighborhood of x, such that |zx-h(y)|<ε for all yU. Choose yUA such that f(y)<f(x)-losc(f,x)+ε. Then (55)hy-fyzx-ε-fx-loscf,x+ε=zx-fx+loscf,x-2ε=L-2ε.Since ε>0 is arbitrary, we get that d(f,h)L and the proof of the claim is over.

Corollary 16.

Let X be a normal space, fRX, and AX be a discrete closed subset. For each xA fix zxRX and consider the affine subspace (56)Z=hCX:hx=zx  for  all  xA.Then (57)df,Z=max12oscf,Mwhere (58)M=supxAmaxzx-fx+loscf,x,fx+uoscf,x-zx=supxAmaxzx-f¯x,zx-f_x.

Proof.

By Theorem 15 we have that d(f,Z)max{1/2osc(f),M}. The other inequality is true by the claim of the proof of Theorem 15.

Corollary 17.

Let X be a normal space, fRX, and AX be a discrete closed subset. For each xA fix (59)zxf¯x-12oscf,f_x+12oscf.Then there is gC(X) such that g(x)=zx for all xA and (60)df,g=12oscf.

Proof.

If xA and zx[f¯(x)-1/2osc(f),f_(x)+1/2osc(f)], we have that (61)maxzx-f¯x,zx-f_x12oscfso by Theorem 15 there is a function gC(X) such that g(x)=zx and (62)df,g=max12oscf,M=12oscf.

The proofs of Theorems 12 and 15 are very similar. In fact we can combine both theorems.

Theorem 18.

Let X be a normal space, fRX, AX be a discrete closed subset, and B be a closed subset of X such that f is continuous at x for all xB. For each xA fix zxRX. Then there is gC(X) such that g(x)=zx for all xA, g(x)=f(x) for all xBA, and (63)df,g=max12oscf,Mwhere (64)M=supxAmaxzx-fx+loscf,x,fx+uoscf,x-zx=supxAmaxzx-f¯x,zx-f_x.

We omit the proof because it is very similar to the proofs of Theorems 12 and 15. If we read the proof of Theorem 15, we have to change the definition of gi by (65)gix=zxif  xA,fxif  xBAhixif  xAB,and then we get by Lemma 11  gC(X) that g1gg2, and then we have to check that g is the desired function.

Data Availability

All our findings are theoretical ones and are included within the article.

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

The author is supported by the Spanish grants MTM2017-83262-C2-2-P of the Spanish Ministry of Economy and 20906/PI/18 of Fundación Séneca.

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