J Ascel (1965, p241,246,196 ,281,286): in his' lectures on Functional equations and their applications)' pictures attached
on page (196) Azcel, Describes a condition for a real valued function as intern: if\forall x>y,in [A,B] F(x,y)\s in [x,y] and F(x,y) is in (y,x) for all y
Dear William,
F is a function of two variables, which are extremities of an interval. Without obstacles we can reformulate:
F* is a function that associates a real number to a compact interval, F*([x,y]) = F(x,y).
Internality of F on [a,b] can be perceived as follows:
the value associated by F* to any closed subinterval [x,y] of [a,b] lies in (x,y).
The internality of F guarantees stability.
In French, a similar (somewhat weaker) property is:
Soit g une fonction définie sur un intervalle I. On dit que I est stable par g si et seulement si g(I) est un sous-ensemble de I.
@Wiliam,
can you please quote literally? If you have the book of János Aczél, scane please the corresponding page and include it to your replay. Thanks.
Dear Viera Thanks for this. I think that is the best idea. I will do so.
dear Viera i attached some of the pages, page 196, and 241 in particular. Sometimes it just looks like a mixture of strict monotonic and conditional surjectivity/continuity. ; asserting the exiseting of order respecting function midpoints. On other occassion its looks like something by saying , almost what those values are that F(x)/2 +F(y)/2 not only exists, if F(x) F(y) do x, y such if that x
Dear William,
F is a function of two variables, which are extremities of an interval. Without obstacles we can reformulate:
F* is a function that associates a real number to a compact interval, F*([x,y]) = F(x,y).
Internality of F on [a,b] can be perceived as follows:
the value associated by F* to any closed subinterval [x,y] of [a,b] lies in (x,y).
The internality of F guarantees stability.
In French, a similar (somewhat weaker) property is:
Soit g une fonction définie sur un intervalle I. On dit que I est stable par g si et seulement si g(I) est un sous-ensemble de I.
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