5.29. On the set of continuous functions Cla, b] consider the metrics PM and p defined by PM(S.8) max,rela,b}llf(x)- g(x)], and P(f.g)= If(x)-g(x)| dx. These metrics were introduced in Exercise 5.8 and Example 5.5, respectively. (a) Use Theorem 5.15 to prove that the topology induced by pM on Cla, b] is finer than the topology induced by p. (b) Show that for every c,cz > 0 there exists f e Cla, b) such that maxreta.bill/(x)|]= and |Ifoldx = c2. Chapter 5 Metric Spaces (c) Let Z e Cla, b] be the function defined by Z(x) Given e > (Hint: Part (b) helps.) (d) What does Theorem 5.15 allow us to conclude from (c)? O for all x e [a, b). 0, show that no 8 0 exists such that B,(Z. 8) C Bo(Z. E).