Exercise 7.1 (1) One observes that, for any choice of the function ?, the sequence satisfies all the conditions in Definition 7.1.2, and therefore it is convergent to ? ?0 in . (2) If ? is non-identically zero, the sequence, although uniformly convergent to 0, is not convergent in because it does not satisfy the condition (ii) in the definition 7.1.2. Indeed, in this case, there exists at least one such that ?'(t) ?0. Let t _k =t/k for , and let us observe that for every , which shows that the sequence of the first-order derivatives does not converge uniformly to 0 on . (3) If ? is non-identically zero, the sequence, although uniformly convergent to 0, it is not convergent in because it does not satisfy the condition (i) in Definition 7.1.2. Indeed, in this case, there exists at least one such that ?(t) ?0. Let us observe that the term of rank k of the unbounded sequence lies in the support of the term of the same rank of the sequence of functions considered.

Exercise 7.2 (1) (t)=2 ?(t). (2) (t)=cost 2 ?(t) ?sin t sgn (t)=2 ?(t) ? sin t sgn (t). (3) (t)=t ?(t)+sgn (t)=sgn (t). (4) (t)=t ?( t ?1)+sgn ( t ?1)= ?(