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math-2011-note2 - 2.7 Differentiability and Continuity A...

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2.7 Differentiability and Continuity A function f is differentiable at x 0 if the following limit exists lim h 0 f ( x 0 + h ) - f ( x 0 ) h . A function f is continuous at x 0 if lim h 0 f ( x 0 + h ) exists and lim h 0 f ( x 0 + h ) = f ( x 0 ) . Example 1 (Non-differentiable, discontinuous functions) . f ( x ) = | x | , g ( x ) = braceleftBigg x + 1 , if x 0 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 One-Variable 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 0 and f prime ( x 0 ) > 0. This means that lim h 0 f ( x 0 + h ) - f ( x 0 ) h = f prime ( x 0 ) > 0 For small and positive h , we have f ( x 0 + h ) - f ( x 0 ) > 0 . If we write x 1 for x 0 + h , we have x 1 > x 0 f ( x 1 ) > f ( x 0 ) for x 1 near x 0 . This means that f is an increasing function near x 0 . Ex. Show that x 1 < x 0 f ( x 1 ) < f ( x 0 ) for x 1 near x 0 , if f prime ( x 0 ) > 0. Ex. Show that f is a decreasing function near x 0 , if f prime ( x 0 ) < 0.
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