the further mentioned are T. Fine 'On the existence of quantitative probability';

Roberts 'Note on FInes axioms' and Roberts: Correction 'Notes on FInes axioms;

As well as Narens (1985, 2007, 2000) on dedekind completeness ; denumerable density, density and dedekind completeness (i presume denumerable density just means first order countable and separable, countable order dense set, not a dense total  ordering (bottomless, between-ness,density,non-atomic,and connected), as in the rational in the reals

Apparently there is a distinction between continuity (pointwise?) and cauchy continuity; ie

uniform continuity (compact and dedekind complete)\subset\,

dedekekind continuity(strong continuity? as in rao and rao)

subset\,cauchy continuity'

\subset'\,continuous'(pointwise?)

subset(pointwise?)\subset, sequentially continuous ; I presume that strongly non-atomic/convex ranged spaces are either uniformly continuous or slightly stronger but not necessarily absolutely continuous spaces

monotone continuity does not really fit in perhaps, but  if, so perhaps,  'a subset of the sequentially continuous' (or identical to them), or a super-set of the continuous orders/function. In between in strength between continuity and sequential continuity, although apparently now continuity itself means something rather weak (i presumed that it, point-wise continuity, and cauchy continuity were roughly the same). The notion of weak continuity has also been mentioned before

The Requirements of of the Dedekind form(strong continuity as in rao and rao?) are satisfication of the basic qualitative probability axioms of a weak order, that is

A. non negative, non-triviality, additivity/scott axiom/some other condition etc plus (plus villegas' monotone continuity (or maybe fines weaker condition),archimededean requirements/sigma algebras if they are not already implict in below, in the case of qp space)

and which satisfies the further constraints below B.

of B.1. (no holes):dedekind completeness-least upper bound property in reals  Thist is sometimes considered, "cauchy completeness' plus an archimedean requirement.

B.2. connected-ness (a total ordering), A>=B or A

This is a good question.

To get started, consider

Strongly continuous:  A function f : X → Y is strongly continuous (strongly

θ-continuous) at x ∈ X if given any open set V in Y containing f(x), there exists an

open set U in X containing x such that f(U) ⊆ V.

https://fada.birzeit.edu/bitstream/20.500.11889/4047/1/On%20--Continuity%20And%20Strong%20--Continuity%20(1).pdf

Thanks for this; I get the impression that sometimes these terms are used in different contexts; A lot of the time, the word continuity in the (qualitative probability literaturre/prob philosophy)/expected utitlity economics/decision theory literature refers to some kind of sequential, monotone continuity or relatively strong arch-imedean property).

They do not often mean point-wise continuity or cauchy continuity although apparently there is a difference; which is generally what is meant by plain 'continuity'. Although the latter is apparently stronger. I get the impression when I look at the definitions that

-strong continuity refers to cauchy continuity (which is not quite uniformly continuous but stronger than 'continuity); see rao rao 'theory of finite charges' page 142'

-weak continuity- oftens refers to point-wise albeit not cauchy continuous functions (if there are such); which is still stronger then sequential (and I think monotone continuity).

 I have also come across uniform strong continuity (see Alon Lehrer 2014)

Denition 2. A set P of probability measures is uniformly strongly continuous if:

(a) For any event B, u(B) > 0 if and only if u'(B) > 0, for every pair of probabilities

; u,u'\inP.

(b) For every epsilon> 0, there exists a finite partition {G1,Gj,...Gr}of S, such that for all j,(Gj) < epsilon for all u \in P.

It appears to be combination of Savage's fine-ness, standard non-atomicity without the tightness constraint with an archimeadean or monotone continuity like constraint but without sigma algebras or the other way around)

Which is presumably stronger than strong continuity, as in rao and rao (1983), plus a further constraint. I presume this is weaker then uniform continuity (ie uniformly strong, not strongly uniform) but I might be wrong).

Although sometimes, they appear refer to a non-atomic/bottomless/dense and total order; that satisfies the basic constraints (non triviality, non negativity, scotts condition)on a boolean algebra (such as the rational in the reals); ie

'The combination of 'total order (or complete weak order), no gaps (non atomic, or dense order) plus an arch-imedean constraint, non triviality, non,negativity,' ;'cancellation/additivity, and perhaps separability (and countable order dense subset-first countable space at least) is often classified as continuous:

.  This constraint is generally weaker than the least upper bound property/perhaps a bounded-ness requirement;

And it (appears to be) definitely weaker then monotone continuity plus the sigma algebra joint constraints-This is clearly weaker then both notions above (cauchy/pointwise continiuity) although possibly stronger (in some sense) than sequential continuity) There may be gaping holes, even if there are no gaps.

The latter two when combined with non-atomicity, and total order, and his other basic constraints entail convex-ranged-ness/strongly atomicity. 

I presume that strong non-atomicity/convex-rangedness roughly approximates uniform continuity or something slightly stronger

see rao and rao page 142 for their definition of strong continuity; It may be a topological or qualtiatitve notion, (likewise with the definitions of continuity and uniformly strongly continuous, and the stronger variants of archimeanity, archimedean continuity, strong archimedean continuity, and the stronger version of monotone continuity introduced by Gilboa and Marinacci

ee page 50 and 51 in the Vind book for the reference to gaplessness M1 and an archimedean M2 property entailing continuity given the basic axioms he lists as well.

. I am not sure what the difference is between deriving point-wise continuity (as opposed to Cauchy continuity);

I presume that neither case must compactness be required (or that sets, sequences etc are closed, or contain their limit points).

Which would entail uniform continuity.

The other two papers (i notice, are Econometrica, Vol. 78, No. 2 (March, 2010), 755–770

OBJECTIVE AND SUBJECTIVE RATIONALITY IN

A MULTIPLE PRIOR MODEL

ITZHAK GILBOA' and differentated ambiguity and ambiguity attitude 2004) which give definitions of continuity, strong monotone and archi-medean continuity.It appears to me that most people think of Cauchy continuity when continuity or pointwise continuity is mentioned.

Although I get the impression that suitably bounded rationals may be continuous (and more strongly continuous than 'monotone or sequentially continuous' but not cauchy continuous')

I presume that in the first case it does not even need to be dedekeind complete, just atomless (dense) and bounded, whilst in the former it must be bounded complete (as to whether any further kind of requirement or archimedeanity is required for dedekind completeness and whether this is a stronger form of continuity  again is I guess a different story;

There are various notions of dede-kind and cauch-y completeness; where the latter is weaker and does mean that it requires that is its least upper bound is in real valued. I would presume that for the most part they both must be bounded and not just complete. seehttps://en.wikipedia.org/wiki/Least-upper-bound_property

and also https://en.wikipedia.org/wiki/Cauchy-continuous_function

'Every uniformly continuous function is also Cauchy-continuous, and any Cauchy-continuous function is continuous. Conversely, if X is totally bounded, then every Cauchy-continuous function is uniformly continuous and if the space X is complete, then every continuous function on X is Cauchy-continuous too. More generally, even if X is not complete, as long as Y is complete, then any Cauchy-continuous function from X to Y can be extended to a function defined on the Cauchy completion of X; this extension is necessarily unique.'

Examples and non-examples; I presume the differences are only apparent in metrif and topological spaces and not in real numbers; although it alludes to the fact that a continuous  (but not cauchy continuous) process/function need not be complete

HI James, Thanks for this I was wondering which book or text you got that reference from. I am trying to find a maths stack exchange question where someone asked also about the difference between weak and strong (it was implied that strongly continuous, pointwise/standard continuous/cauchy continuous appear to be one and the same thing, whilst what passes as weakly continuous or continuous is not continous at all; such as the rationals)

See also https://books.google.com.au/books?id=mTNQvfe54CoC&pg=PA145&lpg=PA145&dq=strong+continuity+rao+definition&source=bl&ots=fUPaAQya9M&sig=XTJ2xM5fXCk4QArf_64gxVxyPoI&hl=en&sa=X&ved=0ahUKEwiFnaCtjsvSAhXCj1QKHXs3AIEQ6AEIRjAG#v=onepage&q=strong%20continuity%20rao%20definition&f=false

I was wondering if you know of an example of a function or plot that satisfies

(1)say pointwise or weak continuous function but not cauchy continuous/strong continuous ; i presume that this, and the further weaker sequential continuity/monotone continuitous 'function' (if that makes sense) would just be a whole lot scattered disconnected points which would (or at least the latter) would look continuous from first inspection, when really it is hardly at all

(2)(4) equicontinuous

(2) uniformly strong continuous but not uniformly continuous

(3) versus a pair of functions that are uniformly equicontinuous

(5) holder continuous

and whether there are versions in betweeen ( stronger form of uniform continuity that is not holder continuous

Hi James see http://math.stackexchange.com/questions/197338/examples-of-a-strongly-continuous-function-and-a-weakly-continuous-function

http://math.stackexchange.com/questions/235782/about-weakly-continuous-functions

http://math.stackexchange.com/questions/199738/weakly-continuous-function-but-not-strongly-continuous?rq=1

I am wondering whether whether either of these if relates to point-wise continuity at all points, versus cauchy continuous.

see the first four commentsin 8/examples-of-a-strongly-continuous-function-and-a-weakly-continuous-function

hanks for the answer. I am still confused. Some people say that strongly continuous functions include f(x)=a, based on the following definition: f is a strongly continuous function if and only if there exists a δ>0 for all ϵ>0 such that for all real z, if 0