Let F(x) := x cos(Ï/x) for x [0; 1] and F(0) := 0. It will be seen that f := Fʹ R*[0, 1] but that

Let F(x) := x cos(Ï€/x) for x ˆˆ [0; 1] and F(0) := 0. It will be seen that f := Fʹ ˆˆ R*[0, 1] but that its absolute value |f| = |Fʹ| ˆ‰ R*[0, 1]. (Here f(0) := 0.)
(a) Show that Fʹ and |Fʹ| are continuous on any interval [c, 1], 0 (b) If ak := 2/(2k + 1) and bk := 1/k for k ˆˆ N, then the intervals [ak, bk] are non overlapping and 1/k

01 1. k=1