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4.8: Continuity on Compact Sets. Uniform Continuity

This page is a draft and is under active development. 

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I. Some additional important theorems apply to functions that are continuous on a compact set (see §6).

Theorem 4.8.1

If a function f:A(T,ρ),A(S,ρ), is relatively continuous on a compact set BA, then f[B] is a compact set in (T,ρ). Briefly,

the continuous image of a compact set is compact.

Proof

To show that f[B] is compact, we take any sequence {ym}f[B] and prove that it clusters at some qf[B].

As ymf[B],ym=f(xm) for some xm in B. We pick such an xmB for each ym, thus obtaining a sequence {xm}B with

f(xm)=ym,m=1,2,

Now by the assumed compactness of B, the sequence {xm} must cluster at some pB. Thus it has a subsequence xmkp. As pB, the function f is relatively continuous at p over B (by assumption). Hence by the sequential criterion (§2),xmkp implies f(xmk)f(p); i.e.,

ymkf(p)f[B].

Thus q=f(p) is the desired cluster point of {ym}.

This theorem can be used to prove the compactness of various sets.

Example 4.8.1

(1) A closed line segment L[¯a,¯b] in En( and in other normed spaces ) is compact, for, by definition,

L[¯a,¯b]={¯a+tu|0t1}, where u=¯b¯a.

Thus L[¯a,¯b] is the image of the compact interval [0,1]E1 under the mapf:E1En, given by f(t)=¯a+tu, which is continuous by Theorem 3 of §3. (Why?)

(2) The closed solid ellipsoid in E3,

{(x,y,z)|x2a2+y2b2+z2c21},

is compact, being the image of a compact globe under a suitable continuous map. The details are left to the reader as an exercise.

lemma 4.8.1

Every nonvoid compact set FE1 has a maximum and a minimum.

Proof

By Theorems 2 and 3 of §6, F is closed and bounded. Thus F has an infimum and a supremum in E1 (by the completeness axiom), say, p=infF and q=supF. It remains to show that p,qF.

Assume the opposite, say, qF. Then by properties of suprema, each globe Gq(δ)=(qδ,q+δ) contains some xB (specifically, qδ<x<q) other than q( for qB, while xB). Thus

(δ>0)FG¬q(δ);

i.e., F clusters at q and hence must contain q (being closed). However, since qF, this is the desired contradiction, and the lemma is proved.

The next theorem has many important applications in analysis.

Theorem 4.8.2

(Weierstrass).

(i) If a function f:A(T,ρ) is relatively continuous on a compact set BA, then f is bounded on B; i.e., f[B] is bounded.

(ii) If, in addition, B and f is real (f:AE1), then f[B] has a maximum and a minimum; i.e., f attains a largest and a least value at some points of B.

Proof

Indeed, by Theorem 1,f[B] is compact, so it is bounded, as claimed in (i).

If further B and f is real, then f[B] is a nonvoid compact set in E1, so by Lemma 1, it has a maximum and a minimum in E1. Thus all is proved.

Note 1. This and the other theorems of this section hold, in particular, if B is a closed interval in En or a closed globe in En( or Cn) (because these sets are compact - see the examples in §6). This may fail, however, if B is not compact, e.g., if B=(¯a,¯b). For a counterexample, see Problem 11 in Chapter 3, §13.

Theorem 4.8.3

If a function f:A(T,ρ),A(S,ρ), is relatively continuous on a compact set BA and is one to one on B (i.e., when restricted to B), then its inverse, f1, is continuout on f[B].

Proof

To show that f1 is continuous at each point qf[B], we apply the sequential criterion (Theorem 1 in §2). Thus we fix a sequence {ym}f[B],ymqf[B], and prove that f1(ym)f1(q).

Let f1(ym)=xm and f1(q)=p so that

ym=f(xm),q=f(p), and xm,pB.

We have to show that xmp, i.e., that

(ε>0)(k)(m>k)ρ(xm,p)<ε.

Seeking a contradiction, suppose this fails, i.e., its negation holds. Then (see Chapter 1, §§1–3) there is an ϵ>0 such that

(k)(mk>k)ρ(xmk,p)ε,

where we write “mk” for “m” to stress that the mk may be different for different k. Thus by (1), we fix some mk for each k so that (1) holds, choosing step by step,

mk+1>mk,k=1,2,

Then the xmk form a subsequence of {xm}, and the corresponding ymk=f(xmk) form a subsequence of {ym}. Henceforth, for brevity, let {xm} and {ym} themselves denote these two subsequences. Then as before, xmB,ym=f(xm)f[B], and ymq,q=f(p). Also,by(1),

(m)ρ(xm,p)ε(xm stands for xmk).

Now as {xm}B and B is compact, {xm} has a (sub)subsequence

xmip for some pB.

As f is relatively continuous on B, this implies

f(xmi)=ymif(p)

However, the subsequence {ymi} must have the same limit as {ym}, i.e., f(p). Thus f(p)=f(p) whence p=p (for f is one to one on B), so xmip=p.

This contradicts (2), however, and thus the proof is complete.

Example 4.8.2

(3) For a fixed nN, define f:[0,+)E1 by

f(x)=xn.

Then f is one to one (strictly increasing) and continuous (being a monomial; see §3). Thus by Theorem 3, f1 (the nth root function) is relatively continuous on each interval

f=[an,bn].

hence on [0,+).

See also Example (a) in §6 and Problem 1 below.

II. Uniform Continuity. If f is relatively continuous on B, then by definition,

(ε>0)(pB)(δ>0)(xBGp(δ))ρ(f(x),f(p))<ε.

Here, in general, δ depends on both ϵ and p (see Problem 4 in §1); that is, given ϵ>0, some values of δ may fit a given p but fail (3) for other points.

It may occur, however, that one and the same δ (depending on ϵ only) satisfies (3) for all pB simultaneously, so that we have the stronger formula

(ε>0)(δ>0)(p,xB|ρ(x,p)<δ)ρ(f(x),f(p))<ε.

Definition

If (4) is true, we say that f is uniformly continuous on B.

Clearly, this implies (3), but the converse fails.

Theorem 4.8.4

If a function f:A(T,ρ),A(S,ρ), is relatively continuous on a compact set BA, then f is also uniformly continuous on B.

Proof

(by contradiction). Suppose f is relatively continuous on B, but (4) fails. Then there is an ϵ>0 such that

(δ>0)(p,xB)ρ(x,p)<δ, and  yet ρ(f(x),f(p))ε;

here p and x on δ. We fix such an ϵ and let
δ=1,12,,1m, Then for each δ (i.e., each m), we get two points xm,pmB with ρ(xm,pm)<1m and ρ(f(xm),f(pm))ε,m=1,2, Thus we obtain two sequences, {xm} and {pm}, in B. As B is compact, {xm} has a subsequence xmkq(qB). For simplicity, let it be {xm} itself; thus xmq,qB.

Hence by (5), it easily follows that also pmq (because ρ(xm,pm)0. By the assumed relative continuity of f on B, it follows that

f(xm)f(q) and f(pm)f(q) in (T,ρ).

This, in turn, implies that ρ(f(xm),f(pm))0, which is impossible, in view of (6). This contradiction completes the proof.

Example 4.8.1

(a) A function f:A(T,ρ),A(S,ρ), ic called a contraction map (on A) iff

ρ(x,y)ρ(f(x),f(y)) for all x,yA.

Any such map is uniformly continuous on A. In fact, given ε>0, we simply take δ=ε. Then x,pA

ρ(x,p)<δ implies ρ(f(x),f(p))ρ(x,p)<δ=ε,

as required in (3).

(b) As a special case, consider the absolute value map (norm map) given by

f(¯x)=|¯x| on En( or another normed space ).

It is uniformly continuous onEn because

||¯x||¯p|||¯x¯p|, i.e., ρ(f(¯x),f(¯p))ρ(¯x,¯p),

which shows that f is a contraction map, so Example (a) applies.

(c) Other examples of contraction maps are

(1) constant maps (see §1, Example (a)) and

(2) projection maps (see the proof of Theorem 3 in §3).

Verify!

(d) Define f:E1E1 by

f(x)=sinx

By elementary trigonometry, |sinx||x|. Thus (x,pE1)

|f(x)f(p)|=|sinxsinp|=2|sin12(xp)cos12(x+p)|2|sin12(xp)|212|xp|=|xp|,

and f is a contraction map again. Hence the sine function is uniformly continuous on E1; similarly for the cosine function.

(e) Given A(S,ρ), define f:SE1 by

f(x)=ρ(x,A) where ρ(x,A)=infyAρ(x,y)

It is easy to show that

(x,pS)ρ(x,A)ρ(x,p)+ρ(p,A)

i.e.,

f(x)ρ(p,x)+f(p), or f(x)f(p)ρ(p,x)

Similarly, f(p)f(x)ρ(p,x). Thus

|f(x)f(p)|ρ(p,x)

i.e., f is uniformly continuous (being a contraction map).

(f) The identity map f:(S,ρ)(S,ρ), given by

f(x)=x

is uniformly continuous on S since

ρ(f(x),f(p))=ρ(x,p) (a contraction map!) 

However, even relative continuity could fail if the metric in the domain space S were not the same as in S when regarded as the range space (e.g., make ρ discrete!)

(g) Define f:E1E1 by

f(x)=a+bx(b0).

Then

(x,pE1)|f(x)f(p)|=|b||xp|;

i.e.,

ρ(f(x),f(p))=|b|ρ(x,p).

Thus, given ε>0, take δ=ε/|b|. Then

ρ(x,p)<δρ(f(x),f(p))=|b|ρ(x,p)<|b|δ=ε,

proving uniform continuity.

(h) Let

f(x)=1x on B=(0,+).

Then f is continuous on B, but not uniformly so. Indeed, we can prove the negation of (4), i.e.

(ε>0)(δ>0)(x,pB)ρ(x,p)<δ and ρ(f(x),f(p))ε.

Take ε=1 and anyδ>0. We look for x,p such that

|xp|<δ and |f(x)f(p)|ε,

i.e.,

|1x1p|1,

This is achieved by taking

p=min(δ,12),x=p2.( Verify! )

Thus (4) fails on B=(0,+), yet it holds on [a,+) for any a>0.
(Verify!)


This page titled 4.8: Continuity on Compact Sets. Uniform Continuity is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Elias Zakon (The Trilla Group (support by Saylor Foundation)) via source content that was edited to the style and standards of the LibreTexts platform.

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