hw23solutions - f is continuous. 2. Let f,g be dierentiable...

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Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: April 5, 2010 Assignment 23 1. A differentiable function f : I R is said to be uniformly differentiable on I := [ a,b ] if for every ε > 0 there exists δ > 0 such that if 0 < | x - y | < δ and x,y I then ± ± ± f ( x ) - f ( y ) x - y - f 0 ( x ) ± ± ± < ε. Show that if f is uniformly differentiable on I , then f 0 is continuous on I . Fix ε > 0, and let x,z I so that 0 < | x - z | < δ with δ as in the definition of unifomrly differentiable. Then, | f 0 ( x ) - f 0 ( z ) | ≤ ± ± ± f ( x ) - f ( z ) x - z - f 0 ( x ) ± ± ± + ± ± ± f ( x ) - f ( z ) x - z - f 0 ( z ) ± ± ± < 2 ε. Thus we see that
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Unformatted text preview: f is continuous. 2. Let f,g be dierentiable on R and suppose that f (0) = g (0) and f ( x ) g ( x ) for all x 0. Show that f ( x ) g ( x ) for all x 0. Set h ( x ) = g ( x )-f ( x ). Then, by the Mean Value Theorem, for any x 0 there is a y (0 ,x ) so that g ( x )-f ( x ) = h ( x )-h (0) = h ( y )( x-0) = ( g ( y )-f ( y )) x . 1...
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