Prove that f is integrable on [0, 2] by the method in example 36.4

Example 36.4. Define a function by f(x) = 1 if x 0. Set 6 = =/4. Let P = (To, . ..; In) be any partition of [0, 2] such that | |P|| l(mi - mi) + f(s;)(1; - j-1) + _ 2(x - mi-1)Recall RS(f, P, S) -31 = Etif(s )(ci -zi1). Let us consider two cases: Case 1: Assume m; * 1 for all i. Then there is a j such that Ij1I(x - mi)+f(s;)(2; -1 1)+ ) 2(1 -1-1) i=1 i j+1 = (xj_1 - co) + f(s;)(x; - 1;-1) + 2(1n -1;) = (cj_1 -0) + f(s;)(x; - 1;_1) + 2(2-1;) Since my 1 1(mi -mi)+f(s;)(2; - j 1) + f(sji)(oil -;) + ) 2(1 -mi-1) 1= 1 i=j+2 = (*j-1 -0) + f(s;)(1 -=;_1) + f(s;+1)(@;+1 - 1) + 2(2-1;+1) In a similar way to the first case we get (1 -6) +2(1 -6)