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Just e
Exercise 6.2.6. Assume fn → f on a set A. Theorem 6.2.6 is an example of a typical type of question which asks whether a trait possessed by each fn is inherited by the limit function. Provide an example to show that all of the following propositions are false if the convergence is only assumed to be pointwise on A. Then go back and decide which are true under the stronger hypothesis of uniform convergence. (a) If each fn is uniformly continuous, then f is uniformly continuous (b) If each fn is bounded, then f is bounded (c) If each fn has a finite number of discontinuities, then f has a finite number of discontinuities. (d) If each fn has fewer than M discontinuities (where M ENis fixed), then f has fewer than M discontinuities. (e) If each fn has at most a countable number of discontinuities, then f has at most a countable number of discontinuities.
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