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MAT371defs4 - Chapter 4 1 Three equivalent definitions of...

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Chapter 4. 1. Three equivalent definitions of differentiable and derivative : Let f : D R with x 0 an accumulation of D , and x 0 D . The following are three are equivalent definitions of differentiable: (a) The limit L := lim x x 0 f ( x ) - f ( x 0 ) x - x 0 exists. (b) The limit L := lim t 0 f ( x 0 + t ) - f ( x 0 ) t exists. (c) There exists a number L and a function φ such that lim t 0 φ ( t ) = 0 and f ( x 0 + t ) = f ( x 0 ) + Lt + ( t ) , t ∈ { t | t + x 0 D } The limit L , if it exists, is called the derivative of f at x 0 and is denoted by f 0 ( x 0 ). If a function has a derivative at each x S then we say the function is differentiable on S . If a function is differentiable on its domain, we simply say that the function is differentiable. 2. Let f : D R , x 0 D such that there exist a neighborhood Q of x 0 where x Q D we have f ( x ) ( resp. ) f ( x 0 ), then we say that f has a relative minimum
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