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Problem 7 Let a be a monotonically increasing function on [0,1] such that (0) < (1). Prove that the Dirichlet function 1 , x rational R, x - , x irrational is not RS-integrable with respect to an increasing a : [0,1] R. Problem 8 Let a : [a,b] R be a monotonically increasing function, let f : [a, b] R, x C for some real number c. Prove that f is Riemann-Stieltjes integrable and that b C da = c(a(b)-a(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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Real Analysis Problems
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