Exam2018Proposed solution included

MAT 302 ANALYSIS II Exam 2018

MAT 302: End of Semester Examination for MAT302 Analysis II, Department of Mathematics, Faculty of Science, University of Buea, 27 July 2018 (Instructor: Dr. Ndambomve P.). Exercise 1: definitions of core metric-space notions (Lipschitz function, fixed point, convergent/Cauchy sequence, completeness, continuity, uniform continuity, pointwise/uniform convergence, compactness) and statements of the Contraction Mapping Principle, Weierstrass M-Test and Uniform Cauchy Criterion. Exercise 2: proving every Lipschitz map is uniformly continuous, a non-expansive-but-fixed-point-free map on R, continuity preserving compactness, and closed subsets of compact sets being compact. Exercise 3: pointwise/uniform convergence analysis of three sequences/series of functions, including a Weierstrass M-Test application.

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University of Buea

Faculty of Science - FS

Mathematics - MAT

MAT3022018

ANALYSIS II

Exam · Second Semester · Level 300

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1.
Carefully define the following notions in the framework of metric spaces: a) Lipschitz function, b) fixed point of a map, c) convergent sequence, d) Cauchy sequence, e) Complete metric space, f) continuous functions, g) uniformly continuous function, h) pointwise convergence of sequences of functions, i) uniform convergence of sequences of functions, j) compact set. (1x8 marks)
1.
Prove that every Lipschitz map is a uniformly continuous function. (3 marks)