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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 ∀ ε > ∃ δ x,ε > 0 : 0 <  x y  < δ x,ε ⇒ fl fl fl fl f ( x ) f ( y ) x y k fl fl fl fl < ε. Equivalently ∀ ε > ∃ δ x,ε > 0 : 0 <  δ  < δ x,ε ⇒ fl fl fl fl f ( x + δ ) f ( x ) δ k fl fl fl fl < ε. 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 δ x, 1 > 0 such that fl fl fl f ( x ) f ( y ) x y k fl fl fl < 1 whenever 0 <  x y  < δ x, 1 (because we can take ε = 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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 Spring '09
 Calculus, Derivative, Continuous function, dx, limy→x

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