midtermsoln

# midtermsoln - Solution to Mat125A Midterm Exam Instructor...

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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 > 0 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 > 0 such that | f ( x ) - L | ≤ = 1 whenever x > M . That is L - 1 f ( x ) L + 1 (3) whenever x M . On the other hand, since f is continuous on [0 , M ], there exists x 1 , x 2 [0 , M

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