# 9 how is the notion of closed set defined 10 what

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9. How is the notion of ’closed set’ defined?? 10. What does it mean to say a sequence of measurable functions <f n > converges to a function f in measure?

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MAA5616/FINAL EXAM/PART B Autumn, 1997 Page 3 of 4 11. Let E be a non-empty subset of , and suppose that f:E is a function. What does it mean to say f is continuous at a point x ε E ?? 12. Suppose that <f n > is a sequence of real-valued functions defined on a non-empty set E and f is a real-valued function defined on E. What does it mean to say the sequence <f n > converges pointwise to f on E ?? 13. Suppose that <f n > is a sequence of real-valued functions defined on a non-empty set E and f is a real-valued function defined on E. What does it mean to say the sequence <f n > converges uniformly to f on E ?? 14. Suppose that f:E is a function with E . What does it mean to say f is uniformly continuous on E ?? 15. How is the Lebesgue outer measure of a subset E of the real line defined in terms of the length of an interval l(I)??
MAA5616/FINAL EXAM/PART B Autumn, 1997 Page 4 of 4 16. How do we define the measurability of a subset E of the real line? 17. Suppose that A is a subset of the real line. What does it mean to say a function f:A is measurable?? 18. Let f:[a,b] be a function. What does it mean to say f is of bounded variation on [a,b] ?? 19. What does it mean to say something is true almost everywhere?? 20. Let f:[a,b] be a function. What does it mean to say f is absolutely continuous on [a,b] ??
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• Continuous function, Limit of a sequence, Topological space

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