02: A) If y (x) < M on [a,b] and the restriction of f(x) to every...
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00 Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
Please answer it step by step and Question 2. uniformly converge is defined by *f=0* clear handwritten, please, also, beware that for the x you have 2 conditions , such as x>n and 0<=x<=n 1- For all n > 1 define fn: [0, 1] → R as follows: (i if n!x is an integer 10 otherwise Prove that fn + f pointwise where f:[0,1] → R is defined by ſo if x is irrational f(x) = 3 11 if x...
Integral: If you know all about it you should be easy to prove..... Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information to prove thatL(f)≤L(g)andU(f)≤U(g). Information: g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0 g is discontinuous at every rational number in[0,1]. g is Riemann integrable on [0,1] based...
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...
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...
13.1.11. Problem. Let f(x) = x and g(x) = 0 for all x ∈ [0,1]. Find a function h in B([0,1]) such that du(f,h) = du(f,g) = du(g,h). (3 problems) 13.2.6. Problem. Given in each of the following is the nth term of a sequence of real valued functions defined on (0, 1]. Which of these converge pointwise on (0, 1]? For which is the convergence uniform? (a) a z" (b) z+ nr. (c) a+ re-na 13.2.7. Problem. Given in...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
Consider f : [0, 1] x [0, 1] C R2 + R defined by f(x,y) = ſi if y is rational 2x if y is irrational Show that f is not integrable over R by the following steps: in (a) For each n > 1, find a Sn:= Eosi,jan f(a 6? b., in [0, 1] for 0 < i, j < n such that the Riemann sum converges as n + 0.[10 pts] n 1 n2 n i, ja (b)...
Let f : [0, 1] x [0, 1] + R be defined by f(x, y) = {1 if y = 23, 0 if y + x2 Show that f is integrable on (0, 1] x [0, 1]. You may take the previous problem as given
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...