Problem 7 Let a be a monotonically increasing function on [0,1] such that (0) < (1).
Prove that the Dirichlet function
1 , x rational
, x irrational
is not RS-integrable with respect to an increasing a : [0,1]
Problem 8 Let a : [a,b] R be a monotonically increasing function, let f : [a, b]
for some real number c. Prove that f is Riemann-Stieltjes integrable and that
C da = c(a(b)-a(a)) .
Problem 9 Suppose f,g : [a,b] R are two functions such that
f (x) < g(x) for all . € [a,b].
Further assume that O is a monotonically increasing function on [a,b].
(a) Given a partition P of [a, b], prove that L(f,P,) < L(g, P, ).
(b) Deduce from (a) that jof da < jog b da.
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