2.

(a) Prove that for each n E N, Pn := {~ : j = 0,1, ... ,n}

is a partition of [0,1].

(b) Prove that a bounded function f is integrable on [0, 1] if

(*) fo:= lim L

(f, Pn ) = lim U

(f, Pn ), n---+OCl n---+oo in which case J01 f(x) dx equals f o.

(c) For each ofthe following functions, use Exercise 1, p. 17, to find formulas for the upper and lower sums of f on Pn , and use them to compute the value of J01 f(x) dx.

(a)

((J)

f(x) = x.

f(x) = { ~ ° ~ x < 1/2 1/2 ~ x ~ 1.