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Analysis2010jan-revised

# Analysis2010jan-revised - g n on R such that f n converges...

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January 2010 Preliminary Exam in Analysis. 1. Let X be a connected metric space. Given two points p, q X and a number ² > 0, prove that there exist an integer n > 0 and points a 0 , a 1 , . . . , a n X such that a 0 = p , a n = q , and d ( a j , a j - 1 ) < ² for all j = 1 , 2 , . . . , n. 2. Suppose that f : (0 , 1] R is a bounded continuous function such that for every t R the set { x (0 , 1]: f ( x ) = t } is finite. Prove that f is uniformly continuous on (0 , 1]. 3. Prove or disprove the following: if a function f : ( - 1 , 1) R is differentiable on ( - 1 , 1) and f 0 (0) = 0, then for every δ > 0 there exists ² > 0 such that fl fl fl fl f ( t ) - f ( s ) t - s fl fl fl fl < δ whenever - ² < s < t < ². 4. Let f be a bounded real-valued function on [ a, b ] with a discontinuity at c ( a, b ). Let α ( x ) be monotonically increasing on [ a, b ] with α ( c - ) < α ( c ) < α ( c +). Prove that f is not Riemann-Stieltjes integrable with respect to α on [ a, b ]. 5. Give examples of sequences of functions { f n } and
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Unformatted text preview: { g n } on R such that { f n } converges uniformly, { g n } converges uniformly but { f n g n } does not converge uniformly on R . 6. Let φ,ψ : R 3 → R be continuously diﬀerentiable functions and deﬁne F : R 3 → R 3 by F ( x,y,z ) = ( φ ( x,y,z ) ,ψ ( x,y,z ) ,φ 2 ( x,y,z ) + ψ 2 ( x,y,z )) (a) Check whether or not the inverse function theorem applies to F at any point ( x ,y ,z ), i.e., check if F satisﬁes the hypothesis of the inverse function theorem at any point ( x ,y ,z ). (b) Suppose that F ( ~a ) = ~ b for some points ~a, ~ b ∈ R 3 . Explain geometrically why F does not have an inverse function from an open set V ⊂ R 3 containing ~ b to an open set U ⊂ R 3 containing ~a ....
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