Does Uniform Continuity Imply Preseveration of Openedd

Sets, Functions and Metric Spaces

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

Remark 14.4

The difference between the concepts of continuity and uniform continuity concerns two aspects:

(a)

uniform continuity is a property of a function on a set, whereas continuity is defined for a function in a single point;

(b)

$\delta$ participating in the definition (14.50) of continuity, is a function of $\epsilon$ and a point p, that is, $\delta =\delta \left(\epsilon ,p\right)$ , whereas $\delta$ , participating in the definition (14.17) of the uniform continuity, is a function of $\epsilon$ only serving for all points of a set (space) $\mathcal{X}$ , that is $\delta =\delta \left(\epsilon \right)$ .

Evidently, any uniformly continued function is continuous but not inverse. The next theorem shows when both concepts coincide.

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INTEGRAL OPERATORS

L.V. KANTOROVICH , G.P. AKILOV , in Functional Analysis (Second Edition), 1982

THEOREM 5.

Suppose that the kernel of an integral operator is almost uniformly continuous with respect to τ for τ = τ₀ and suppose that the conditions of Theorem 3 are satisfied for each τ, where none of the constants (C ₁, C ₂, σ, etc.) appearing in the statement of that theorem depend on x. Then the integral operator U _τ, given by

(20) $\begin{array}{l}{y}_{\tau }=U_{\tau }\left(x\right),y_{\tau }\left(s\right)=\underset{D}{\int }{K}_{\tau }\left(s,t\right)x\left(t\right)dt\\ (x\in {\text{L}}^p\left(D\right),y_{\tau }\in {\text{L}}^q\left(D\right)),\end{array}$

depends continuously on the parameter, in the sense that

$\underset{\tau \to {\tau }_0}{\lim }\left|\right|U_{\tau }-U_{{\tau }_0}\left|\right|=0.$

Proof. We have

$y_{{\tau }_0}\left(s\right)-y_{\tau }\left(s\right)=\underset{D}{\int }\left[K_{{\tau }_0}\left(s,t\right)-K_{\tau }\left(s,t\right)\right]x\left(t\right)dt.$

Using the notation of the proof of Theorem 3, we find a bound for the constant

$C_1^{\left(\rho \right)}-\underset{s\in D^{\prime }}{\mathrm{vrai}\sup }{\left[\underset{D}{\int }|K_{{\tau }_0}\left(s,t\right)-K_{\tau }\left(s,t\right){|}^{\rho }dt\right]}^{1/\rho }$

Given ɛ>0, h >0 and s, we determine the δ>0 and A (s) ⊂ D corresponding to these in the definition of almost uniform continuity. Then, assuming for simplicity that ρ ⩾ 1, ^†

$\begin{array}{l}{\left[\underset{D}{\int }|K_{{\tau }_0}\left(s,t\right)-K_{\tau }\left(s,t\right){|}^{\rho }dt\right]}^{1/\rho }\le {\left[\underset{D^{\prime }|A\left(s\right)}{\int }|K_{{\tau }_0}\left(s,t\right)-K_{\tau }\left(s,t\right){|}^{\rho }dt\right]}^{1/\rho }+\\ +{\left[\underset{A\left(s\right)}{\int }|K_{{\tau }_0}\left(s,t\right)-K_{\tau }\left(s,t\right){|}^{\rho }dt\right]}^{1/\rho }\\ \le \epsilon {\left[\mathrm{mes}D\right]}^{1/\rho }+{\left[\underset{A\left(s\right)}{\int }|K_{{\tau }_0}\left(s,t\right)-K_{\tau }\left(s,t\right)|\rho .\frac{p}{\rho }dt\right]}^{1/r}{\left[\underset{A\left(s\right)}{\int }dt\right]}^{1/{\left(\frac{r}{\rho }\right)}^{\prime }\rho }\le \\ \le \epsilon {\left[\mathrm{mes}D\right]}^{1/\rho }+2C_1{h}^{1/\rho }{-}^{1/r}.\end{array}$

Therefore C ^(ρ) ₁⩽ ɛ [mes D]^1/ρ+ 2 C ₁ h ^{1/ρ-1/ r} ; that is, the value of C ^(ρ) ₁ for the operator U _τ0 - U _τ can be made as small as we please. Since, by Theorem 1 (see (5)), we have

$\left|\right|U_{{\tau }_0}-U_{\tau }\left|\right|\le {\left[C_1^{\left(\rho \right)}\right]}^{1-\sigma /q}{\left[2C_2\right]}^{\sigma /q},$

it follows that || U _τ0 - U _τ|| is also as small as we please, which proves the theorem.

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Handbook of Computability Theory

Pour-El Marian Boykan , in Studies in Logic and the Foundations of Mathematics, 1999

3.2.2 Application of the computability structure to prove equivalence of definitions

It now becomes clear that the three types of definitions for computability on C[a, b], the Weierstrass Approximation definition, the definitions based on computable functional, and the definition involving effective uniform continuity (see section 2.3 of Part I) are all equivalent. The sequence 1, x, x ², … is a computable sequence on all of these definitions. (This is not hard to see.) Hence the polynomials with rational coefficients can be arranged in a computable sequence. (See Axiom 1 above.) Since these polynomials are dense in C[a, b], they form an effective generating set. We now apply the Stability Lemma to obtain the desired result.

Analogous results hold for L^p [a, b].

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Nonlinearity and Functional Analysis

In Pure and Applied Mathematics, 1977

2.1A Boundedness and continuity

The map f ∈ M(X, Y) is called continuous (with respect to convergence in norm) if x_n → x in X always implies f(x_n ) → f(x) in Y. f is said to be bounded if it maps bounded sets into bounded sets. f is called locally bounded if each point in the domain of f has a bounded neighborhood N such that f(N) is bounded. In the case when f is linear, the two concepts of continuity and boundedness are equivalent; but this is not true in general.

Since continuous maps of a finite-dimensional Banach space X into a Banach space Y are necessarily bounded, one naturally seeks to extend this result to infinite-dimensional spaces. To accomplish this, we introduce the notion of uniform continuity of the mapping f.

(2.1.1) A mapping f is uniformly continuous if for every ∈ > 0 there exists a δ(∈) > 0 such that ∥x − y∥ < δ implies ∥f(x) – f(y)∥ < ɛ.

Clearly a uniformly continuous mapping is continuous. In fact, we have (2.1.2) A uniformly continuous mapping is bounded.

Proof: It suffices to show that f maps any sphere S_r = {x| ∥x∥ ≤ r} into a bounded set. For any ɛ > 0, by the uniform continuity of f, there is a δ > 0 such that ∥x − y∥ < δ implies ∥f(x) – f(y∥ < ɛ for x,y ∈ S_r. Choose n to be any positive integer satisfying nδ > 2r. Then if a,b ∈ S_r , there are n points x_i ∈ S_r , with ∥x_i − x _i−1∥ < δ and x ₀ = a,x _n−1 = b. Hence

$\left|\right|f\left(a\right)-f\left(b\right)\left|\right|\le \sum _{i=1}^{n-1}\left|\right|f\left(x_i\right)-f\left(x_{i-1}\right)\left|\right|<\left(n-1\right)\in ,$

a number independent of the choice of a and b; from which the result follows.

Actually (besides continuity with respect to convergence in norm) there are three distinct and important notions of (sequential) continuity for mappings f between general Banach spaces X and Y. These notions are obtained by considering the possible actions of f on the weak as well as the strong topologies of X and Y. Thus a map f ∈ M(X, Y) may (i) map strongly convergent sequences in X into weakly convergent sequences in Y, (ii) map weakly convergent sequences in X into weakly convergent sequences in Y, or (iii) map weakly convergent sequences in X into strongly convergent sequences in Y. This last continuity property (iii) is called complete continuity since it implies the other two. Property (ii) is called demicontinuity. The alternative notions of continuity are sometimes useful in proving boundedness of a map f independently of uniform continuity assumptions. In fact, we have

(2.1.3) Let X be a reflexive Banach space and f ∈ M(X, Y). If f maps weakly convergent sequences in X into sequences weakly convergent in Y, then f is bounded.

Proof: We argue by contradiction. Suppose there is a bounded sequence {x_n } in X such that ∥f(x_n )∥ → ∞. By the reflexivity of X, {x_n } has a weakly convergent subsequence {x_nj } (say). By hypothesis, {f(x_nj )} is weakly convergent and hence, by (1.3.11): uniformly bounded. But this fact contradicts the fact ∥f(x_n )∥ → ∞.

In the sequel continuous mappings between Banach spaces X and Y are denoted C(X, Y).

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Elements of Stability Theory

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

Proof

Suppose that such function V (t, x) exists. Show that $\overline{x}\left(t,x_0,t_0\right)\to 0$ as $t\to \infty$ whenever $\Vert x_0\Vert$ is small enough, that is, show that for any $\epsilon >0$ there exists $T=T\left(\epsilon \right)$ such that $\Vert \overline{x}\left(t,x_0,t_0\right)\Vert <\epsilon$ for all $t>T$ . Notice that by the uniform continuity V (t,x) and by Corollary 20.1 all trajectories of $\overline{x}\left(t,x_0,t_0\right)$ (20.1) remain within the region where $\Vert \overline{x}\left(t,x_0,t_0\right)\Vert <\epsilon$ if $\Vert x_0\Vert <\delta$ . So, the function $W\left(t,\overline{x}\left(t,x_0,t_0\right)\right)$ remains bounded too. Suppose that $\Vert \overline{x}\left(t,x_0,t_0\right)\Vert$ does not converge to zero. By monotonicity of $V\left(t,\overline{x}\left(t,x_0,t_0\right)\right)$ , this means that there exists $\in >0$ and a moment $T=T(\in )$ such that for all $t\ge T(\in )$ we have $\Vert \overline{x}\left(t,x_0,t_0\right)\Vert <\epsilon$ . Since W (t, x) is a positive-definite function, it follows that

for all $t\ge T(\in )$ and, hence, by (20.20) we have

$\begin{array}{ll}V\left(t,\overline{x}\left(t,x_0,t_0\right)\right)\hfill & =V\left(t,\overline{x}\left(t,x_0,t_0\right)\right)-{\displaystyle \underset{s=t_0}{\overset{t}{\int }}W\left(s,\overline{x}\left(s,x_0,t_0\right)\right)ds}\hfill \\ \hfill & \le V\left(t,\overline{x}\left(t,x_0,t_0\right)\right)-\alpha t\to -\infty \hfill \end{array}$

which contradicts the condition that V (t, x) is a positive-definite function. The fact that this result is uniform on t ₀ follows from Corollary 20.1. Theorem is proven.

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Uniform Spaces

Eric Schechter , in Handbook of Analysis and Its Foundations, 1997

UNIFORM CONTINUITY

18.7.

Notation. Let u be a uniform space, with uniformity u determined by gauge D. Let ((x _α, x′_α) : α ∈ A) be a net in X×X, and let ε be its eventuality filter on X×X.

Show that the following conditions are equivalent.

(A)

u ⊆ ∈

(B)

For each U ∈ u, we have eventually (x _α, x′_α) ∈ U.

(C)

d(x _α, x′_α) → 0 in ℝ for each d ∈ D. We shall abbreviate this as D(x _α, x′_α) → 0.

We emphasize that the last condition does not say ${\sup }_{d\in D}d\left(x_{\alpha },{x^{\prime }}_{\alpha }\right)\to 0.$

18.8.

Definition. Let X,U and (Y, v) be uniform spaces, and let D and E be any gauges that determine the uniformities U and v, respectively. Let φ : X→ Y be some function. Then the following conditions are equivalent. If any (hence all) of these conditions hold, we say φ is uniformly continuous.

(A)

Whenever V∈ v, then the set

$\begin{array}{lll}{\left(\phi \times \phi \right)}^{-1}\left(V\right)\hfill & =\hfill & \left\{\begin{array}{lll}\left(x,x^{\prime }\right)\in X\times X\hfill & :\hfill & \left(\phi \left(x\right),\phi \left(x^{\prime }\right)\right)\in V\hfill \end{array}\right\}\hfill \end{array}$

is a member of U . That is, the inverse image of each entourage is an entourage. (This is the definition of uniform continuity used in 9.8.)

(B)

Whenever D(x _α, x′_α) → 0 in X, then E(φ(x _α), φ(x″_α)) → 0 in Y. (Notation is as in 18.7(C).)

(C)

For each number ε > 0 and each pseudometric e∈ E, there exist some number δ > 0 and some finite set D′ ⊆ D such that

$\begin{array}{lll}\underset{d\in D^{\prime }}{\max }d\left(x_1,x_2\right)<\delta \hfill & \Rightarrow \hfill & e\left(\phi \left(x_1\right),\phi \left(x_2\right)\right)<\epsilon .\hfill \end{array}$

(We emphasize that the choice of δ and D′ depends on ε and e but not on x ₁ or x ₂. Compare this with 15.14(D).)

(D)

For each e∈ E, there exists a finite set D _e ⊆ D and a function r _e : [0, +∞) → [0,+∞) that is continuous and increasing, and satisfies γ _e (0) = 0 and

$\begin{array}{lll}e\left(\phi \left(x\right),\phi \left(x^{\prime }\right)\right)\hfill & \le \hfill & {\gamma }_e\left(\underset{d\in D_e}{\max }d\left(x,x^{\prime }\right)\right)\hfill \end{array}.$

Such a system of sets D _e and functions γ _e will be called a modulus of uniform continuity for φ.

Note that if the gauge D is directed (as defined in 4.4.c), then conditions 18.8(C) and 18.8(D) can be simplified slightly: the sets D' and D _c may be taken to be singletons {d}.

18.9.

Examples and related properties

a.

If the uniformity on X is given by a pseudometric d, then sequences suffice in 18.8(B) (regardless of whether Y is pseudometrizable). That is, a mapping φ : X→ Y is uniformly continuous if and only if

$\text{whenever}d\left(x_n,{x^{\prime }}_n\right)\to 0\text{inX, then}E\left(\phi \left(x_n\right),\phi \left({x^{\prime }}_n\right)\right)\to 0\text{ ;in}Y,$

with notation as in 18.7(C).

b.

Any Hölder-continuous function from one metric space into another is uniformly continuous. The converse is false. For instance, define $f:\left[0,e^{-1}\right]\to ℝ$

$\begin{array}{lll}f\left(t\right)\hfill & =\hfill & \{\begin{array}{cc}0& \text{when}t=0\\ -1/\ln t& \text{when}0<t\le e^{-1}\end{array}\hfill \end{array}$

Show that f is not Hölder continuous with any exponent. It is easy to see that f is continuous; then the uniform continuity of f will follow by a compactness argument in 18.21.

c.

Any uniformly continuous function is continuous (where each uniform space is equipped with its uniform topology). This can be proved using uniformities or using gauges; the student is urged to give both proofs.

d.

Show that the function f(t) = 1/t is continuous, but not uniformly continuous, on the open interval (0, 1). Use this fact to give two different metrics on (0, 1) that yield different uniformities but that both yield the usual topology.

e.

(Preview.) Let p: X→ Y be a linear map from one topological vector space to another — or more generally, an additive map from one topological Abelian group to another. Let X and Y be equipped with their usual uniform structures (see 26.37). If p is continuous, then p is uniformly continuous; see 26.36.c.

f.

Let X be a set, let {(Y _λ, E _λ) : λ ∈ ∧} be a collection of gauge spaces, and let φ_λ : X→ Y _λ be some mappings. Show that the initial uniformity on X determined by the φ_λ's and E _λ's (as in 9.16) is equal to the uniformity on X determined by the gauge $D={\displaystyle {\cup }_{\lambda \in \wedge }\left\{e{\phi }_{\lambda }:e\in E_{\lambda }\right\}}$ where $\left(e{\phi }_{\lambda }\right)\left(x,x^{\prime }\right)=e\left({\phi }_{\lambda }\left(x\right),{\phi }_{\lambda }\left(x^{\prime }\right)\right).$ We may call this the initial gauge determined by the φ_λ's and E _λ's (although any other gauge uniformly equivalent to this one will generally do just as well).

An important special case: When $X={\displaystyle {\prod }_{\lambda \in \wedge }{Y}_{\lambda }}$ and the φ_λ's are the coordinate projections, we obtain the product gauge.

g.

The forgetful functor from uniform spaces to topological spaces preserves the formation of initial objects.

That is, the uniform topology (U) determined by an initial uniformity πλ S determined by π_λ's and uniformities v _λ is equal to the initial topology determined by the λ_τ's and the uniform topologies τ(v _λ) determined by those uniformities.

18.10.

Theorem on uniform continuity of extensions. Let X and Y be uniform spaces, let X ₀ ⊆ X be dense, let φ : X→ Y be continuous, and suppose that the restriction of φ to X ₀ is uniformly continuous. Then φ is uniformly continuous on X. In fact, if some gauges are specified for X and Y, then any modulus of uniform continuity for the restriction of φ to X ₀ is also a modulus of uniform continuity for φ on X.

In particular, if φ is continuous and the restriction of φ to X ₀ is Hülder continuous or Lipschitzian, then φ is HüUlder continuous or Lipschitzian with the same constant.

Hints: Use notation as in 18.8(D). Let any x, x′ ∈ X be given. Choose a net ((x _α,x′_α)) in X ₀ × X ₀ that converges to (x, x′). For each α, we have

$\begin{array}{ccc}e\left(\phi \left(x_{\alpha }\right),\phi \left({x^{\prime }}_{\alpha }\right)\right)& \le & \gamma e\left(\underset{d\in D_e}{\max }d\left(x_{\alpha },{x^{\prime }}_{\alpha }\right)\right)\end{array}.$

Holding e fixed, take limits to obtain a corresponding inequality for (x, x′).

18.11.

Characterization of uniformly equivalent gauges. Let D and E be gauges on a set X. Then the following conditions are equivalent:

(A)

D and E are uniformly equivalent — i.e., they generate the same uniformity.

(B)

The identity map i _X : X→ X is uniformly continuous in both directions between the gauge spaces (X, D) and (X, E)

(C)

For each net ((x _α,x′_α) : α ∈ A) in X×X, we have

$\begin{array}{ccc}D\left(x_{\alpha },{x^{\prime }}_{\alpha }\right)\to 0& \iff & E\left(x_{\alpha },{x^{\prime }}_{\alpha }\right)\to 0\end{array}$

with notation as in 18.7(C).

Hint: A uniformity, being a proper filter, is the eventuality filter for some net.

18.12.

Further exercise. Let U be a uniformity on a set X. Then the largest gauge that is compatible with U (as defined in 5.32) is the set of all pseudometrics d: X×X→ [0, +∞) that are jointly uniformly continuous — i.e., uniformly continuous when X×X is given its product uniformity and [0, +∞) is given its usual uniformity. (Compare this with 16.20.)

18.13.

If D is any gauge, then D is uniformly equivalent to its max closure and its sum closure, defined as in 4.4.c.

(Hence it is often possible to replace a gauge with a directed gauge; thus in many contexts we may assume a gauge is directed.)

18.14.

Definition. We shall say β is a bounded remetrization function if:

(i)

β is a mapping from [0, +∞) onto a bounded subset of [0, +∞);

(ii)

β is continuous;

(iii)

β is increasing; that is, s≤ t⇒ β(s) ≤ β(t);

(iv)

β(t) = 0 ⇔ t= 0; and

(v)

β is subadditive; that is, β(s + t) ≤ β(s) + β(t).

Show that

a.

arctan(t), tanh(t), min{1, t}, and t/(1 + t) are bounded remetrization functions of t. (Hint: See 12.25.e.) Note that min{1, t} is not injective.

b.

If β is a bounded remetrization function and d is a (pseudo)metric on a set X, then e(x, y) = β(d(x, y)) defines a (pseudo)metric e= β оd on X that is uniformly equivalent to d and is bounded.

c.

If β is a bounded remetrization function and D is a gauge on a set X, then {β о d: d∈ D} is a gauge on X that is uniformly equivalent to D and is uniformly bounded — i.e., we have sup{β(d(x, y)) : x, y ∈ X, d ∈ D} < ∞.

18.15.

Example. The usual metric on ℝ is d(x, y) = |x - y|. Another metric, bounded and uniformly equivalent to the usual one, is e(x, y) = arctan(|x- y|). On the other hand, ρ(x, y) = | arctan(x) – arctan(y)| is a bounded metric on ℝ that is equivalent, but not uniformly equivalent, to the usual metric. (All three metrics yield the same topology.)

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Basic Representation Theory of Groups and Algebras

In Pure and Applied Mathematics, 1988

Proof

Consider the upward directed set F of all functions f in L₊ (X × Y) such that f ≤ Ch _U ; and put ϕ_f(x) = ${\displaystyle \int }$ f(x, y)d|v|y (f ∈ F, x ∈ X ). An easy uniform continuity argument shows that ϕ _f is continuous, hence in L(X); and by 9.8 an 9.10(2)

(5) $\begin{array}{cc}{\displaystyle \int {\varphi }_{f}d\left|\mu \right|={\displaystyle \int fd\left|\mu \times v\right|\le k<\infty }}& \left(f\in F\right)\end{array}$

where k = |μ × v|_e (U).

Now for each positive integer n define

$V_n=\left\{x\in X:{\varphi }_f\left(x\right)>n^{-1}\text{for}\text{some}\text{f}\text{in}F\right\}.$

Since the ${\varphi }_f$ are continuous, V_n is an open subset of U ₁ ; and since supp(v) = Y, $\cup$ _n V_n = U ₁. We shall show that V_n ∈ ℰ _μ. Indeed, for each f in F and each pair of positive integers, n, m, let $D_{f,m}^n$ be the compact set

$\left\{x\in X:{\varphi }_f\left(x\right)\ge \frac{1}{n}+\frac{1}{m}\right\}.$

Then $n^{-1}\left|\mu \right|\left(D_{f,m}^n\right)\le {\displaystyle \int {\varphi }_{f}d\left|\mu \right|;}$ so by (5)

(6) $\left|\mu \right|\left(D_{f,m}^n\right)\le nk.$

Now, for fixed n, the family of all $D_{f,m}^n$ (f ∈ F; m = 1, 2,…) is cofinal in the upward directed set of all compact subsets of V_n . Hence V_n ∈ C _μ by (6) and Remark 1, 8.15.

Since $\cup$ _n V_n = U ₁, we have shown that U ₁ is a countable union of open sets in ℰ _μ Similarly, U ₂ is a countable union of open sets in ℰ_v .

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Selected Topics of Real Analysis

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

Theorem 16.10. (First sufficient (Riemann's) condition)

Assume that α ↑ on [a, b]. If for any ε > 0 there exists a partition P_ε of [a, b] such that P_n is finer than P _ε implies

(16.75) $0\le U(P_n,f,\alpha )-L(P_n,f,\alpha )<\epsilon$

then f ∈ R _[a,b] (α).

Proof

Since by α ↑ on [a, b] we have

$U(P_n,f,\alpha )\le S(P_n,f,\alpha )\le L(P_n,f,\alpha )$

In view of (16.75) this means that S (P_n, f, α) has a limit when n → ∞ which, by the definition (15.11), is the Riemann–Stieltjes integral. Theorem is proven.

Theorem 16.11(Second sufficient condition)

If f is continuous on [a, b] and α is of bounded variation on [a, b], then f ∈ R _[a,b] (α).

Proof

Since by (15.55) any α of bounded variation can be represented as α (x) = α⁺ (x) − α⁻ (x) (where α⁺ ↑ on [a, b] and α⁻ ↑ on [a, b]), it suffices to prove the theorem when α ↑ on [a, b] with α (a) < α (b). Continuity of f on [a, b ] implies uniform continuity, so that if ε > 0 is given, we can find δ = δ (ε) > 0 such that |x − y| < δ implies |f (x) − f (y)| < ε/A where A = 2 [α (b) − α (a)]. If P_ε is a partition with the biggest interval less than δ, then any partition P_n finer than P_ε gives

since

$M_i-m_i=\text{sup}\left\{f(x)-f(y):x,y\in [x_{i-1},x_i]\right\}$

Multiplying (16.76) by Δ α _i and summing, we obtain

$U(P_n,f,\alpha )-L(P_n,f,\alpha )\le \epsilon /A{\displaystyle \sum }_{i=1}^n\Delta {\alpha }_i=\frac{\epsilon }{2}<\epsilon$

So, Riemann's condition (16.75) holds. Theorem is proven.

Corollary 16.4

For the special case of the Riemann integral when α (x) = x Theorem 16.11 together with (15.23) state that each of the following conditions is sufficient for the existence of the Riemann integral ${\displaystyle {\int }_{x=a}^{b}f(x)dx}$ :

1.

f is continuous on [a, b];

2.

f is of bounded variation on [a, b].

The following theorem represents the criterion (the necessary and sufficient condition) for the Riemann integrability.

Theorem 16.12. (Lebesgue's criterion for integrability)

Let f be defined and bounded on [a, b]. Then it is the Riemann integrable on [a, b], which is f ∈ R _[a,b] (x), if and only if f is continuous almost everywhere on [a, b].

Proof

Necessity can be proven by contradiction assuming that the set of discontinuity has a nonzero measure and demonstrating that in this case f is not integrable. Sufficiency can be proven by demonstrating that Riemann's condition (16.75) (when α (x) = x) is satisfied assuming that the discontinuity points have measure zero. The detailed proof can be found in Apostol (1974).

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Computational Methods for Modelling of Nonlinear Systems

In Mathematics in Science and Engineering, 2007

Example 2

[26] Let X be the space of sequences

$x=\left({\xi }_1,{\xi }_2,\dots \right)$

with ξ → 0 and set

$\Vert x\Vert =max\left|{\xi }_k\right|$

Suppose that ω ⊂ X consists of all sequences of the form

$X_m=\left\{x={\xi }_1,\dots {\xi }_m,0,0,\dots )\right\}$

where σ_i is + 1 or −1. The set ω belongs to the unit ball of the space X and is closed.

We define a functional f on ω; by

$f\left(y\right)={\sigma }_1{\sigma }_2\dots {\sigma }_k$

Since we have

$\Vert y^{\prime }-y^{″}\Vert \ge 1$

for distinct points y′, y″ ∈ ω;, the functional f is uniformly continuous. It is easily extended to the unit ball in X with preservation of uniform continuity.

Let

$X_m=\left\{x={\xi }_1,\dots {\xi }_m,0,0,\dots )\right\}$

be a subspace of X. We make no distinction between this space and the space $l\frac{}{m}$ . The restriction of a polynomial functional p_n of degree n on X_m is a polynomial in m variables of degree not greater than n. Therefore if

${\Omega }_m\subset \Omega \cap X_m$

is the set of vertices of the M –dimensional cube [−1, 1] ^m and if f_m is the restriction of the functional f to ω_m, then for any polynomial p_n of degree n and for any m, the inequality

$sup\left\{\left|f\left(x\right)-p_n\left(x\right)\right|forx\in \Omega \right\}\ge E_n\left(f_m\right)$

holds, where E_n (f_m ) is the best uniform approximation of f_m by polynomials in m variables of degree n.

The function f_m is odd with respect to each argument, and in the set of all best-approximation polynomials of this function, there also exists an odd polynomial with respect to each argument. In particular, for m > n the zero polynomial is the best approximation polynomial, since this is the unique polynomial of degree n of m variables which is odd with respect to all arguments, and therefore

$E_n\left(f_m\right)=1$

for m > n.

Thus for any polynomial function p_n, we have

$sup\left\{\left|f\left(x\right)-p_n\left(x\right)\right|forx\in \Omega \right\}\ge 1.$

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On Some Approximation Problems in Topology

A.N. Dranishnikov , in 10 Mathematical Essays on Approximation in Analysis and Topology, 2005

1.1 Large scale topology (Large vs Small)

The classical topology is a small scale science in a sense that it deals with the continuity which is defined in the small scale terms. To see that we restrict ourself to the case of metric spaces. We recall the classical ∈ – δ definition of a uniformly continuous map f : X → Y between metric spaces (X, d_X ) and (Y, d_Y ):

${\forall }_{\in }>\exists \delta >0:d_X\left(x,x^{\prime }\right)<\delta \Rightarrow d_Y\left(f\left(x\right),f\left(x^{\prime }\right)\right)<\in .$

In this definition we assume that ∈ and δ are small numbers. Here is a large scale analog of the uniform continuity:

$\forall \delta >\infty {\exists }_{\in }<\infty :d_X\left(x,x^{\prime }\right)<\delta \Rightarrow d_Y\left(f\left(x\right),f\left(x^{\prime }\right)\right)<\in .$

Here we assume that ∈ and δ are large.

Perhaps for some readers this analogy does not look very analogous because of the difference in the order of quantifiers: We use ∀ _∈ ∃δ in the first case and ∀δ∃ _∈ in the second. To make the analogy visual we reformulate these definitions as follows.

Small scale (classic): A map f : X → Y is uniformly continuous if there exists a function ρ : ℝ₊ → ℝ₊ with lim_t→0 ρ(t) = 0 such that

$d_Y\left(f\left(x\right),f\left(x^{\prime }\right)\right)<\rho \left(d_X\left(x,x^{\prime }\right)\right)forallx,x^{\prime }\in X.$

Large scale: A map f : X →Y is uniformly continuous if there exists a function ρ : ℝ₊ → ℝ₊ with lim_t→0 ρ(t)   =   ∞ such that

$d_Y\left(f\left(x\right),f\left(x^{\prime }\right)\right)<\rho \left(d_X\left(x,x^{\prime }\right)\right)forallx,x^{\prime }\in X.$

By analogy we can say that the large scale topology must be a discipline about the large scale continuity. To define it in the most general setting one should introduce new structure generalizing metric in the large scale direction similarly as the topology structure does it in the small scale aspect. The idea of large scale continuity is presented in the above definition of large scale uniformly continuous maps between metric spaces. For the purpose of our paper it suffices to have only this notion.

Some seeds of the large scale topology can be found in [22], [33], [34], [7], [32], [40], [36] and others.

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Source: https://www.sciencedirect.com/topics/mathematics/uniform-continuity

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