All,
Ok. I got it wrong.
Berlinsk's account of the proof first establishes a function for the distance between the chord and function itself, h.
Then it says,
"Two facts about h must now be invoked. First, h is continuous on (a, b) and second, h is differentiable on (a, b). "
I think my point has to be (if I have one at all) that these two "facts", in combination with a definition of a mean, and the defintion of a slope at a point, are sufficient to entail directly the mean value theorem.
But I dont want to get too hung up on this narrow point. The main point, in my mind, is to figure out the extent to which mathematicians indentify "mathematics" with the formalisms (algebra, etc.) and the extent to which you indentify it with the premisses from which it precedes. .
Anyway, even I perceive that I am starting to beome tiresome.
thanks, all,
Nick
Nicholas S. Thompson
Research Associate, Redfish Group, Santa Fe, NM (nick at redfish.com)
Professor of Psychology and Ethology, Clark University (nthompson at clarku.edu)
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