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 f, 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 hypothe sis of uniform conver

(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 N is 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