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a_ exists absolute value bers bounded function bounded set contain other points contain points containing a_ contains continuous functions continuous on a,b Corollary define a function defined for integral dependent dependent variable desired property different real numbers discontinuous end points endless number Example exists an interval finite number following theorem frontier number function f(x function is bounded function is defined greatest lower bound greatest number Hence Lim f(x incommensurable increment independent variable integers integral values interval a,b interval containing a_ interval f(x irrational numbers jf(x least upper bound Lemma Let f(x Lim f limit as x means of deciding negative neighborhood of a_ noint non-negative number greater ooint point set polynomial positive number problem Proof Properties of Continuous rational numbers real root say f(x say the set second characteristic set is bounded set of numbers set of points statement terval Theorem 6.2 true