# Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal

###### Question:

Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo f Exercise 26: Let ()nEN be a sequence of functions, a, bR. Assume fn is integrable for all n e N and fn → f uniformly on [a,히 for some f : [a,히 → R. Show that f is integrable as well Exercise 27: (Riemann's definition) Let P I be a partition of a, b. A tagged partition (Pckis a partition where we addtitionally choose a point ck in each interval I. Given the tagged partition (P, we define the corresponding Riemann sum by R(f, P) = Σ f(ck)141 k= 1 Riemann's definition of the integral f : [a,시 → R bounded is integrable with J f A if and only if for all ε > 0 there exists δ > 0 such that for any tagged partition (P ak) with Ikl

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