> On Feb 3, 3:48 am, "T.H. Ray" <thray...@aol.com> > wrote: > > Ostap Bender wrote > > > > > > > > > I don't know about that, but to me, a > function is > > > an > > > > > assignment of > > > > > properties to the members of your set, like > the > > > > > weather report: > > > > > > > New York -> 30 degrees, snow > > > > > San Francisco -> 55 degrees, rain > > > > > Los Angeles -> 69 degrees, sunny > > > > > > > etc. > > > > > > Ah, now I see the difficulty. No--the set of > > > cities > > > > with corresponding temperature and weather > > > conditions > > > > are not defined by a function until you define > a > > > > relation among them. A simple relation to > define > > > the > > > > function is the season of the year; If these > > > initial > > > > conditions exist in winter, then in summer this > set > > > > of numbers will change at some particular time > to > > > > something like > > > > > I was talking about the instantaneous weather > report > > > on the Weather > > > Channel or CNN: At this moment, the temperatures > are: > > > > > New York -> 30 degrees, snow > > > San Francisco -> 55 degrees, rain > > > Los Angeles -> 69 degrees, sunny > > > > > > New York, 78 degrees, sunny > > > > San Francisco, 65 degrees, rain > > > > Los Angeles, 85 degrees, partly cloudy > > > > > > A function necessarily transforms... > > > > > What do you mean by "transforms"? Does the > weather > > > report transform > > > the set of cities into a set of temperatures? I > would > > > use the term > > > "assigns" instead of "transforms". > > > > No. The weather report is the result of the > function > > that transforms one set of data into another. It > is > > certainly a transformation, not an assignment of > values, > > because it is not arbitrary. > > > > > > There do exist arbitrary > > assigments of values that are functions; e.g., > ordering > > some group of people by height. Even in such > cases, > > however, transformation applies--the order function > > transforms a random distribution of values into an > > ordered sequence. > > > > > > ... one set of data to > > > > another. The above is an example of a > continuous > > > > function; > > > > > That would depend on your domain. If your domain > is > > > the Earth' surface > > > - then the current temperature is continuous. if > your > > > domain is the > > > set of 100 big cities - there can be no notion of > > > continuity. > > > > The domain for climate is, of course, the earth's > > surface and atmospheres--what else would it be? > > > > Many things. For example, the set of weather stations > that measure the > outside temperature. Any set can serve as the domain > for a function. > Sure. However, the function does not exist without defining a relation between sets. Simply assigning properties to a set does not confer properties of a function. A function is always characterized by a transformation; i.e., a map, a relation.
> > > > The set > > of cities is most certainly embedded in that > surface > > and their climate (by the fixed point theorem)is a > > function of changes in that surface and atmosphere. > > > > Isn't a restricted function still a function in its > own right? > > http://en.wikipedia.org/wiki/Function_(mathematics) > > More precisely, if ? is a function from a X to Y, and > S is any subset > of X, the restriction of ? to S is the function ?|S > from S to Y such > that ?|S(s) = ?(s) for all s in S > > > I don't know what you're getting at. "Restricted" in this context does not obviate transformation (map, relation).
> > Brouwer, who gave us the fixed point theorem, also > gave > > us the deeper result: all real functions are > continuous. > > > > Excuse me? What do you mean by "real functions"? > Functions in the range of the real numbers.