Mathematics 108, Introduction to Differential Calculus ... |
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Page 2
... bers into two parts as follows : whose squares are less than two , in the other we put all remaining rationals . Between these two parts there is no rational frontier number ; part , nor least one in the second part . ( incommensurable ) ...
... bers into two parts as follows : whose squares are less than two , in the other we put all remaining rationals . Between these two parts there is no rational frontier number ; part , nor least one in the second part . ( incommensurable ) ...
Page 4
... bers every one of which is greater than a and less than b . That is , if c is any number in this interval than a < c < b . Notice that the numbers a and b themselves are not in the interval , b- al is called the length of the interval ...
... bers every one of which is greater than a and less than b . That is , if c is any number in this interval than a < c < b . Notice that the numbers a and b themselves are not in the interval , b- al is called the length of the interval ...
Page 6
... bers except 3. The statement · " f ( x ) 2 X = X · 9 3 if x 3 f ( x ) = π if x = 3 " defines another function the range of x being the set of all numbers . If a function is defined by a formula and the range of the independent variable ...
... bers except 3. The statement · " f ( x ) 2 X = X · 9 3 if x 3 f ( x ) = π if x = 3 " defines another function the range of x being the set of all numbers . If a function is defined by a formula and the range of the independent variable ...
Common terms and phrases
absolute value bers bounded function bounded set bounds we say consider contain other points contain points containing a contains continuous functions continuous on a,b Corollary define a function defined for integral denote dependent discontinuous dy dx end points endless number Example Exercises exists an interval finite number frontier number function f(x function is bounded function is defined greatest lower bound greatest number h₁(x h₂(x Hence f(x incommensurable increment integers integral values interval a,b interval containing interval f(x irrational numbers least upper bound Lemma Let f(x Lima limit as x Limx means of deciding negative non-negative pair of points polynomial positive number problem Proof Properties of Continuous range rational numbers say f(x say the set second characteristic set is bounded set is infinite set of numbers set of points statement terval Theorem 6.2 Theorem 9.2 true values of f(x ха