1. Define the function sgn by: ifx>0 ifx=0 sgn(x) = 0 Now define h(x): [0,1]R by...
4. Define f(z) ={z. (Lia z, İftE [0, 1] rational; -z, if x [0,1 irrational 1 f(x) = if z E (, i] rational; Prove that the function f is not integrable on
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I = [0, 1] and compute the value of f du
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I...
work step by step. Thanks
الم 3. Let k : (0,1] x [0, 1] + R be a continuous function and let f be a Lebesgue integrable function on (0,1). (a) Show that for each y € (0,1), 2 + f(-x){}(2", y) is Lebesgue integrable on (0,1). (b) Define g : [0, 1] +R by 8(u) = Sam Slam)x(x, y)dır. 10,11 Prove that g is continuous at cach y € (0, 1].
(h) Define f : [0, 2] + R by 122 if 0 <<<1 f(x) = { ifl<152 Using the limit definition of the derivative and the sequence definition of the limit prove that f'(1) does not exist.
(4) Define the function f : R -> R* by ,--1/2 f(x) x< 0. +oo, |(a) Prove that f is measurable (with respect to the Lebesgue measurable sets). (b) Prove that f is integrable on I 0, 1and compute the value of = f du
(4) Define the function f : R -> R* by ,--1/2 f(x) x
1 Fix an integer N > 1, and consider the function f : [0,1] - R defined as follows: if 2 € (0,1) and there is an integer n with 1 <n<N such that nx € Z, choose n with this property as small as possible, and set f(x) := otherwise set f(x):= 0. Show that f is integrable, and compute Sf. (Hint: a problem from Homework Set 7 may be very useful for 0 this!)
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and compute the value of f du
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
3) (Assessment Topic: Continuity) Consider the function sgn(x)- (a) What is the domain of sgn(x)? (b) Prove that lim sgn(x)メ1? (Hint : Use a contradictory argument in conjunction with the E-δ method.) (c) Show that for any two positive constants p > 0 and r > 0, where 0<p<r, that lim, ++ee)0, by finding For fixed number q > 1 define the function for each e > 0. (d) Show that lim,-0+ K(t)0. (Hint: Write out a few terms on...
31 (a) If fis integrable, prove that fa is integrable. Hint: Given e>0, let h and k be step functions such that h f k and j (k-h) < ε/M, where M is the maximum value of Ik(x) +h(x)]. Then prove that h and k2 are step functions with h' srsk (we may assume that OShSSk since f is integrable if and only if I is-why?), and that I (k2 - h2) <e. Then apply Theorem 3.3. (b) If fand...