The Computability Of The Zero Point Theorem And The Mean Value Theorem

What computable theory is ultimately about is how do you compute continu-ous functions.Ordinary computability theory first introduces computable partial word function f:(?)?*??*explicitly by means of Turing machines.For defining com-putability on other sets M(rational numbers,finite graphs,etc.)words are used as"names" of elements of M.While a machine still transfers words to words,the user interprets these words as names of elements from the set M.Since the sets ?*and N are only countable,they are not sufficient as sets of names for the uncountable set R of real numbers.A Type-2 machine is a special kind of Turing machine,which extends countablesets to uncountable sets.Real numbers can be represented by infinite sequences,for example by infinite decimal fractions(example:3.14159…is a name of ?).In theory of Tpe-2 machine,infinite sequences are used as names of real numbers,and ma-chines which transfer infinite sequences to infinite sequences are used to compute real function.This paper first introduces the basic theory of type 2 Turing machine,then givesthe application of this theory,discusses the zero point theorem,the mean value theo-rem,Lagrange mean value theorem,cauchy mean value theorem,integral mean value theorem,picard existence uniqueness theorem computability.