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Unformatted text preview: CM221A ANALYSIS I NOTES ON WEEK 8 DIFFERENTIATION Let f be a function defined on an interval I . We say that f ( x ) = k if > x, > 0 : 0 < | x- y | < x, fl fl fl fl f ( x )- f ( y ) x- y- k fl fl fl fl < . Equivalently > x, > 0 : 0 < | | < x, fl fl fl fl f ( x + )- f ( x ) - k fl fl fl fl < . and lim y x f ( x )- f ( y ) x- y = k. The number k is called the derivative of f at the point x . If f ( x ) exists for every point x from the interval, we can consider f as a function of the variable x . This function is called the derivative of f . Another notation for the derivative is d d x f ( x ). It may well happen that the limit does not exist, in which case we say that f is not differentiable at x . The right and left limits lim y x +0 f ( x )- f ( y ) x- y and lim y x- f ( x )- f ( y ) x- y (if they exist) are said to be the right and left derivatives of f at the point x . The function f is differentiable if both the right and left derivative exist and have the same value. If x is an end point of the interval I then one can speak only about one of these derivatives (the other limit does not make sense). Theorem. If the derivative f ( x ) exists then f is continuous at x . Proof. If f is differentiable at x then there exists a number x, 1 > 0 such that fl fl fl f ( x )- f ( y ) x- y- k fl fl fl < 1 whenever 0 < | x- y | < x, 1 (because we can take = 1 in the definition). Since fl fl fl f ( x )- f ( y ) x- y- k fl fl fl 6 fl fl fl f ( x )- f ( y ) x- y...
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This document was uploaded on 01/31/2011.
- Spring '09