8/19/2019 01_19_06h

1/1

Problem Session 2, January 19, 2006

Suppose m ≤ f (x) ≤ M on [a, b ] and that α is increasing on [a, b ]. Provethat

m (α (b) − α (a )) ≤ b

a

f dα ≤ M (α (b) − α (a ))

If f is continuous, nonconstant, and α is strictly increasing, prove that theinequality is strict.

June 2003: Let f be a nonnegative, continuous function on [ a, b ] and let αbe strictly increasing on [ a, b ]. Show that if

b

a f dα = 0, then f = 0 on [a, b ].

June 2003: Let α be given by:

α (x) =0; 0 ≤ x < 12; 1 ≤ x < e5; e ≤ x ≤ π

Either directly or by aid of a theorem, calculate the value of the integral

π

0

x 100 dα (x), and show all the details in your calculation.

June 2003: Let f : [− 1, 1] → R be a bounded function. Let α be given by:α (x) = 0 if x ≤ 0, and α(x) = 1 if x > 0. Prove that f ∈ R (α )[− 1, 1] if andonly if f (0+) = f (0). In this case show that

1

− 1 f dα = f (0).

Jan 2001 Dene g(x) =− 1; x = 0x ; 0 < x < 12 x = 1

Compute 1

0

1x + 3

dg(x).

1