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Unformatted text preview: Solution to Mat125A Midterm Exam Instructor: Qinglan Xia Date: Monday, Nov. 2, 2009 1. Suppose f : A → B and g : B → C are uniformly continuous functions. Prove that the composition function h = g ◦ f : A → C is also uniformly continuous. Answer: For any > 0, since g : B → C is uniformly continuous, there exists an η > such that | g ( a )- g ( b ) | ≤ (1) for any a,b ∈ B with | a- b | ≤ η . Since f : A → B is also uniformly continuous, for the η > 0 constructed as above, there exists a δ > 0 such that | f ( x )- f ( y ) | ≤ η (2) whenever x,y ∈ A with | x- y | ≤ δ . Thus, when | x- y | ≤ δ , by (2) and (1), we have | g ( f ( x ))- g ( f ( y )) | ≤ . Therefore, the composition function g ◦ f is also uniformly continuous. 2. Let f : [0 , ∞ ) → R be a continuous function. If lim x →∞ f ( x ) exists and is finite, then prove that f is bounded. Proof: For = 1, since lim x →∞ f ( x ) = L exists and is finite, there exists an M > such that | f ( x )- L | ≤ = 1 whenever x > M . That is L-...
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This note was uploaded on 03/18/2012 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.
- Fall '07