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Analysis2008aug - 1 such that lim n →∞ Z 1 x k ϕ n x d...

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Analysis Preliminary Exam August, 2008 1. Let f : R 2 R be given by the formula f ( x, y ) = ± x 2 y x 2 + y 2 if ( x, y ) 6 = (0 , 0) 0 if ( x, y ) = (0 , 0) . (a) Show that f is continuous at (0 , 0). (b) Prove that the ﬁrst order partial derivatives of f at (0 , 0) exist. (c) Prove that f is not diﬀerentiable at (0 , 0). 2. Suppose f : R R is a continuous function satisfying the equation | f ( x ) - f ( y ) | > | x - y | for all x, y R . Prove that f ( R ) = R . 3. Suppose the boundary of a set in R 2 is a graph of a function. Prove that the function is continuous. 4. Prove or give a counterexample: Let f : (0 , 1) R and g : (0 , 1) R be continuously diﬀerentiable; that is, f, g C 1 (0 , 1). Suppose that lim x 0+ f ( x ) = lim x 0+ g ( x ) = 0 and g and g 0 never vanish on (0 , 1). If lim x 0+ f ( x ) g ( x ) = c for some c R , then lim x 0+ f 0 ( x ) g 0 ( x ) = c. 5. Let { ϕ n } n =1 be a sequence of non-negative Riemann integrable functions on [0

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Unformatted text preview: , 1] such that lim n →∞ Z 1 x k ϕ n ( x ) d x exists for k = 0 , 1 , 2 , ... Show that the limit lim n →∞ Z 1 f ( x ) ϕ n ( x ) d x exists for every continuous function f on [0 , 1]. 1 6. For n = 1 , 2 , 3 , .. , let f n ( x ) = ± 1 if x ∈ { 1 , 1 2 , ..., 1 n } otherwise. (a) Does the sequence { f n } ∞ n =1 converge uniformly on R ? Justify your answer. (b) Assume that α : R → R is an increasing continuous function, prove or disprove the following identity lim n →∞ Z 1-1 f n ( x ) d α ( x ) = Z 1-1 lim n →∞ f n ( x ) d α ( x ) . 2...
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Analysis2008aug - 1 such that lim n →∞ Z 1 x k ϕ n x d...

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