Ma28_S08_Exam2

Ma28_S08_Exam2 - C(b Show that g is continuous at every...

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Math 28 Spring 2008: Exam 2 Instructions: Each problem is scored out of 10 points for a total of 40 points. You may not use any outside materials(eg. notes or books). You have 50 minutes to complete this exam. Problem 1. (a) Let f : A R where A R . State the deﬁnition for f to be uniformly continuous on A . (b) Which of the following functions are uniformly continuous on [0 , )? (i) f ( x ) = sin( x 2 ) (ii) f ( x ) = 1 x +1 Problem 2. Let C be the Cantor set on [0 , 1] obtained in the standard way by successively removing the middle third of each interval. Deﬁne g : [0 , 1] R by g ( x ) = ( 1 x C 0 x 6∈ C. (a) Show that g is discontinuous at every point in
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Unformatted text preview: C . (b) Show that g is continuous at every point not in C . Problem 3. (a) State the Generalized Mean Value Theorem. (b) Let f : R → R be a diﬀerentiable function and suppose that f is bounded. Show that f is uniformly continuous. Problem 4. (a) State the deﬁnition for a function f : A → R to be diﬀerentiable on an interval A . (b) Let f : [-1 , 1] → R be the function deﬁned by f ( x ) = ( x 2 sin 1 x 2 x 6 = 0 x = 0 . Show that f is diﬀerentiable, but that its derivative is unbounded. 1...
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