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Unformatted text preview: 2.7 Differentiability and Continuity • A function f is differentiable at x if the following limit exists lim h → f ( x + h ) f ( x ) h . • A function f is continuous at x if lim h → f ( x + h ) exists and lim h → f ( x + h ) = f ( x ) . Example 1 (Nondifferentiable, discontinuous functions) . f ( x ) =  x  , g ( x ) = braceleftBigg x + 1 , if x ≥ x 2 1 , otherwise • If f is a differentiable function (at all x ), then its derivative f prime ( x ) is another function of x . Definition 2 (p.32) . If the derivative of f , f prime ( x ) is continuous, then we say that f is continuously differentiable , or C 1 . 3 OneVariable Calculus: Applications (ch.3) 3.1 Use of the First Derivative Claim Positive derivative implies increasing function. • Suppose that f is continuously differentiable at x and f prime ( x ) > 0. • This means that lim h → f ( x + h ) f ( x ) h = f prime ( x ) > • For small and positive h , we have f ( x + h ) f ( x ) > . • If we write x 1 for x + h , we have x 1 > x ⇒ f ( x 1 ) > f ( x ) for x 1 near x . • This means that f is an increasing function near x . Ex. Show that x 1 < x ⇒ f ( x 1 ) < f ( x ) for x 1 near x , if f prime ( x ) > 0....
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This note was uploaded on 03/26/2012 for the course ECON 205 taught by Professor Mr.lee during the Spring '11 term at Korea University.
 Spring '11
 Mr.Lee

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