[permalink] [id link]

* Every Lipschitz continuous map is uniformly continuous, and hence a fortiori continuous.

from
Wikipedia

## Some Related Sentences

Every and Lipschitz

__Every__contraction mapping

**is**

__Lipschitz__

**continuous**

**and**

**hence**

**uniformly**

**continuous**( for

**a**

__Lipschitz__

**continuous**function

**,**the constant k

**is**no longer necessarily less than 1 ).

__Every__special

**uniformly**

**continuous**real-valued function defined on the metric space

**is**

**uniformly**approximable by means of

__Lipschitz__functions

**.**

Every and continuous

*****

__Every__

__continuous__functor on

**a**small-complete category which satisfies the appropriate solution set condition has

**a**left-adjoint ( the Freyd adjoint functor theorem ).

__Every__character

**is**automatically

__continuous__from A to C

**,**since the kernel of

**a**character

**is**

**a**maximal ideal

**,**which

**is**closed

**.**

*****

__Every__

__continuous__

**map**from

**a**compact space to

**a**Hausdorff space

**is**closed

**and**proper ( i

**.**e., the pre-image of

**a**compact set

**is**compact

**.**

__Every__

__continuous__

**map**f: X → Y induces an algebra homomorphism C ( f ): C ( Y ) → C ( X ) by the rule C ( f )( φ ) = φ o f for every φ in C ( Y ).

*****

__Every__separable metric space

**is**isometric to

**a**subset of C (), the separable Banach space of

__continuous__functions → R

**,**with the supremum norm

**.**

__Every__topological group can be viewed as

**a**uniform space in two ways ; the left uniformity turns all left multiplications into

**uniformly**

__continuous__maps while the right uniformity turns all right multiplications into

**uniformly**

__continuous__maps

**.**

__Every__

__continuous__function in the function space can be represented as

**a**linear combination of basis functions

**,**just as every vector in

**a**vector space can be represented as

**a**linear combination of basis vectors

**.**

__Every__

**map**that

**is**injective

**,**

__continuous__

**and**either open or closed

**is**an embedding ; however there are also embeddings which are neither open nor closed

**.**

*****

__Every__compact Hausdorff space of weight at most ( see Aleph number )

**is**the

__continuous__image of ( this does not need the continuum hypothesis

**,**but

**is**less interesting in its absence ).

__Every__place south of the Antarctic Circle experiences

**a**period of twenty-four hours '

__continuous__daylight at least once per year

**,**

**and**

**a**period of twenty-four hours '

__continuous__night time at least once per year

**.**

Every and map

__Every__vector v in determines

**a**linear

__map__from R to taking 1 to v

**,**which can be thought of as

**a**Lie algebra homomorphism

**.**

__Every__smooth ( or differentiable )

__map__φ: M → N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:

__Every__inner automorphism

**is**indeed an automorphism of the group G

**,**i

**.**e

**.**it

**is**

**a**bijective

__map__from G to G

**and**it

**is**

**a**homomorphism ; meaning ( xy )< sup >

**a**</ sup >

__Every__algebraic curve C of genus g ≥ 1

**is**associated with an abelian variety J of dimension g

**,**by means of an analytic

__map__of C into J

**.**

__Every__covering

__map__

**is**

**a**semicovering

**,**but semicoverings satisfy the " 2 out of 3 " rule: given

**a**composition of maps of spaces

**,**if two of the maps are semicoverings

**,**then so also

**is**the third

**.**

*****

__Every__bundle of Lie algebras over

**a**smooth manifold defines

**a**Lie algebroid where the Lie bracket

**is**defined pointwise

**and**the anchor

__map__

**is**equal to zero

**.**

*****The embedding theorem for Stein manifolds states the following:

__Every__Stein manifold of complex dimension can be embedded into by

**a**biholomorphic proper

__map__

**.**

__Every__issue of our periodical will therefore include one or more

__map__supplements

**,**

**and**their design will guarantee

**a**

**continuous**

**and**easily accessible supplement in easy-to-manage form with special regard for those who own Stielers Hand-Atlas

**,**Berghaus ’ s Physical Atlas

**,**

**and**other

__map__publications of the ( Perthes ) Institute

**.**

__Every__summer

**,**the town prepares for the one-week summer festival

**,**" Finnsnes i Fest ", aiming to put Finnsnes on the

__map__

**.**

__Every__element x of G gives rise to

**a**tensor-preserving self-conjugate natural transformation via multiplication by x on each representation

**,**

**and**

**hence**one has

**a**

__map__

**.**

Every and is

__Every__legislator from Brasstown Bald to Folkston

__is__going to have his every vote subjected to the closest scrutiny as

**a**test of his political allegiances

**,**not his convictions

**.**

__Every__detail in his interpretation has been beautifully thought out

**,**

**and**of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he

__is__playing with Viennese musicians )

**,**

**and**the gossamer shading throughout

**.**

__Every__taxpayer

__is__well aware of the vast size of our annual defense budget

**and**most of our readers also realize that

**a**large portion of these expenditures go for military electronics

**.**

__Every__few days

**,**in the early morning

**,**as the work progressed

**,**twenty men would appear to push it ahead

**and**to shift the plank foundation that distributed its weight widely on the Rotunda pavement

**,**supported as it

__is__by ancient brick vaulting

**.**

__Every__dream

**,**

**and**this

__is__true of

**a**mental image of any type even though it may be readily interpreted into its equivalent of wakeful thought

**,**

__is__

**a**psychic phenomenon for which no explanation

__is__available

**.**

__Every__library borrower

**,**or at least those whose taste goes beyond the five-cent fiction rentals

**,**knows what it

__is__to hear the librarian say apologetically

**,**`` I'm sorry

**,**but we don't have that book

**.**

__Every__community

**,**if it

__is__alive has

**a**spirit

**,**

**and**that spirit

__is__the center of its unity

**and**identity

**.**

The restricted principle "

__Every__partially ordered set has**a**maximal totally ordered subset "__is__also equivalent to AC over ZF**.**__Every__natural-born citizen of

**a**foreign state who

__is__also an American citizen

**and**every natural-born American citizen who

__is__

**a**citizen of

**a**foreign land owes

**a**double allegiance

**,**one to the United States

**,**

**and**one to his homeland ( in the event of an immigrant becoming

**a**citizen of the US ), or to his adopted land ( in the event of an emigrant natural born citizen of the US becoming

**a**citizen of another nation ).

__Every__line of written text

__is__

**a**mere reflection of references from any of

**a**multitude of traditions

**,**or

**,**as Barthes puts it

**,**" the text

__is__

**a**tissue of quotations drawn from the innumerable centres of culture "; it

__is__never original

**.**

__Every__year

**,**on the last Sunday in April

**,**there

__is__an ice fishing competition in the frozen estuarine waters of the Anadyr River's mouth

**.**

__Every__lattice element of the structure

__is__in its proper place

**,**whether it

__is__

**a**single atom or

**a**molecular grouping

**.**

0.308 seconds.