5. Let f : [a, b] → R be bounded, and a : [a, b] → R monotonically increasing, (a) For a partion P of (a, b), define the upper and lower Riemann-Stieltjes sums with respect to a. (b) (i) Define what it means for f to be Riemann-Stieltjes integrable with respect to a. (ii) State Riemann's Integrability Criterion. (C) Suppose f is both bounded and monotonic, and that a is both monotonically increasing and continuous. Prove that then f...
1. Let a, b E R with a < b and P= {20, 21, ..., In} be a partition of the interval [a, b]. Denote At; = x; – X;-1 for j = 1,2,...,n. Consider a function f : [a, b] → R. (a) (4 points) What do we need to require from f in order to be able to define the upper and lower Riemann sums of f over P? (b) (8 points) Define the upper and the lower...
hint This exercise 5 to use the definition of Riemann integral F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
1. Let a, b E R with a < b and P { To,Ti,. . . ,Tnf be a partition of the interval a, b] Denote ΔΧ,-2 j-rj-1 for J-1, 2, .. . , n. Consider a function f : [a,Ы-R. What do we need to require from f in order to be able to define the upper and lower Riemann sums of f over P?
Let f(x)-12.2-2x-11 and a(z) = x2 + 12n, where Ir] is the largest integer less than or equal to r. (a) Evaluate the upper and lower sums U(f, P) and L(f, P) of f with respect to or if P is the partition {0、름, î,3.3.2) of [O, 2]. 4 42 (b) Explain why f є [0,2] and use results in part (a) to give a range of fda. Let f(x)-12.2-2x-11 and a(z) = x2 + 12n, where Ir] is the...
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
does anyone know how to do this question5b in both direction? 5. Let 01, 02, ... be a strictly increasing sequence in (a,b), and let p > 0 be such that ... Pr = 1. Define a : [a,b] R as follows: a(x) = 0 if a srca, a(x) = p«ifa, <<< an+1, 1 and a(x) = 1 if supan Sasb. (a) For a sc<dsb, describe Aa = o(d) - a(c) in terms of an and Ph. (Thus, convince yourself...
Let n E Z20. Let a, b є R with a < b. Let y-f(x) be a continuous real- valued function on a, b]. Let Ln and R be the left and right Riemann sums for f over a, b) with n subintervals, respectively. Let Mn denote the Midpoint (Riemann) sum for fover la, b with n subintervals (a) Let P-o be a Riemann partition of a,b. Write down a formula for M. Make sure to clearly define any expressions...
Let f : [a, b] → R and xo e (a,b). Assume that f is continuous on [a,b] \{x0} and lim x approaches too x0 f(x) = L (L is finite) exists. Show that f is Riemann integrable. 1. (20 pts) Let f : [a, b] R and to € (a,b). Assume that f is continuous on [a, b]\{ro} and limz-ro f (x) = L (L is finite) exists. Show that f is Riemann integrable. Hint: We split it into...
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 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...