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Re: "natural interval extension"

John et al,

On 08.12.2015 13:25, John Pryce wrote:


A.If one thinks of "interval extension" as being a property of a *function*,
   I maintain that the definition in 1788 is the most "natural
In my opinion, a function has no need for an interval extension. If you want to refer to its range, call it its range (BTW: in several dimensions, even for a continuous function, the range of f:X->R^n on some interval X is not an interval in general). If you want to refer to the (convex, interval, ...) hull of its range, call it the (convex, interval, ...) hull of its range. 


B.If one thinks of it as being a property of an *expression*, then the one
   that's "common in the literature" is sort of natural.
Computing range bounds of functions by evaluating *expressions* is one of the basic tasks of interval arithmetic (and, in my opinion, also one of its greatest accomplishments). While natural interval extensions are redundant for functions, they are essential for expressions. 


Or it would be, if
   the books/articles making the definition made clear that many different
   expressions can define the same function. But they don't. Even Warwick
   Tucker's book, which I find exemplary in most ways, is vague, as Michel
   Hack pointed out on 7 Dec. (WT promised me to make it clearer in a future
   edition.)
The clearest definition I am aware of is given in the book by Ratschek and Rokne, Computer Methods for the Range of Functions (1984):
If f(x) is an expression and X an interval of the same dimension as x, then the (interval-)expression which is obtained by replacing each occurrence of x in f(x) by X is denoted by f(X). The expression f(X) is then called the natural interval extension of f(X) on X. 

Three expressions, no functions involved.


Could we keep the main 1788 text unchanged but insert a footnote about it as a corrigendum?
If we could, I'd strongly support this. We should not pursue a misleading wording, neither in 1788.1 nor in 1788. 

Regards,

Markus

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