4.8: Continuity on Compact Sets. Uniform Continuity
This page is a draft and is under active development.
( \newcommand{\kernel}{\mathrm{null}\,}\)
I. Some additional important theorems apply to functions that are continuous on a compact set (see §6).
If a function f:A→(T,ρ′),A⊆(S,ρ), is relatively continuous on a compact set B⊆A, 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 q∈f[B].
As ym∈f[B],ym=f(xm) for some xm in B. We pick such an xm∈B 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 p∈B. Thus it has a subsequence xmk→p. As p∈B, the function f is relatively continuous at p over B (by assumption). Hence by the sequential criterion (§2),xmk→p implies f(xmk)→f(p); i.e.,
ymk→f(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.
(1) A closed line segment L[¯a,¯b] in En(∗ and in other normed spaces ) is compact, for, by definition,
L[¯a,¯b]={¯a+t→u|0≤t≤1}, where →u=¯b−¯a.
Thus L[¯a,¯b] is the image of the compact interval [0,1]⊆E1 under the mapf:E1→En, given by f(t)=¯a+t→u, which is continuous by Theorem 3 of §3. (Why?)
(2) The closed solid ellipsoid in E3,
{(x,y,z)|x2a2+y2b2+z2c2≤1},
is compact, being the image of a compact globe under a suitable continuous map. The details are left to the reader as an exercise.
Every nonvoid compact set F⊆E1 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,q∈F.
Assume the opposite, say, q∉F. Then by properties of suprema, each globe Gq(δ)=(q−δ,q+δ) contains some x∈B (specifically, q−δ<x<q) other than q( for q∉B, while x∈B). Thus
(∀δ>0)F∩G¬q(δ)≠∅;
i.e., F clusters at q and hence must contain q (being closed). However, since q∉F, this is the desired contradiction, and the lemma is proved. ◻
The next theorem has many important applications in analysis.
(Weierstrass).
(i) If a function f:A→(T,ρ′) is relatively continuous on a compact set B⊆A, then f is bounded on B; i.e., f[B] is bounded.
(ii) If, in addition, B≠∅ and f is real (f:A→E1), 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.
If a function f:A→(T,ρ′),A⊆(S,ρ), is relatively continuous on a compact set B⊆A and is one to one on B (i.e., when restricted to B), then its inverse, f−1, is continuout on f[B].
- Proof
-
To show that f−1 is continuous at each point q∈f[B], we apply the sequential criterion (Theorem 1 in §2). Thus we fix a sequence {ym}⊆f[B],ym→q∈f[B], and prove that f−1(ym)→f−1(q).
Let f−1(ym)=xm and f−1(q)=p so that
ym=f(xm),q=f(p), and xm,p∈B.
We have to show that xm→p, 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, xm∈B,ym=f(xm)∈f[B], and ym→q,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
xmi→p′ for some p′∈B.
As f is relatively continuous on B, this implies
f(xmi)=ymi→f(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 xmi→p′=p.
This contradicts (2), however, and thus the proof is complete. ◻
(3) For a fixed n∈N, 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, f−1 (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)(∀p∈B)(∃δ>0)(∀x∈B∩Gp(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 p∈B simultaneously, so that we have the stronger formula
(∀ε>0)(∃δ>0)(∀p,x∈B|ρ(x,p)<δ)ρ′(f(x),f(p))<ε.
If (4) is true, we say that f is uniformly continuous on B.
Clearly, this implies (3), but the converse fails.
If a function f:A→(T,ρ′),A⊆(S,ρ), is relatively continuous on a compact set B⊂A, 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,x∈B)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
Hence by (5), it easily follows that also pm→q (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. ◻
(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,y∈A.
Any such map is uniformly continuous on A. In fact, given ε>0, we simply take δ=ε. Then ∀x,p∈A
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:E1→E1 by
f(x)=sinx
By elementary trigonometry, |sinx|≤|x|. Thus (∀x,p∈E1)
|f(x)−f(p)|=|sinx−sinp|=2|sin12(x−p)⋅cos12(x+p)|≤2|sin12(x−p)|≤2⋅12|x−p|=|x−p|,
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:S→E1 by
f(x)=ρ(x,A) where ρ(x,A)=infy∈Aρ(x,y)
It is easy to show that
(∀x,p∈S)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:E1→E1 by
f(x)=a+bx(b≠0).
Then
(∀x,p∈E1)|f(x)−f(p)|=|b||x−p|;
i.e.,
ρ(f(x),f(p))=|b|ρ(x,p).
Thus, given ε>0, take δ=ε/|b|. Then
ρ(x,p)<δ⟹ρ(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,p∈B)ρ(x,p)<δ and ρ′(f(x),f(p))≥ε.
Take ε=1 and anyδ>0. We look for x,p such that
|x−p|<δ and |f(x)−f(p)|≥ε,
i.e.,
|1x−1p|≥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!)