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We are given the function f : [0, 4] → R defined by f(x) = 0 for all x # 2 and f(2) = 2. Using the definition of the integral prove that f is (Darboux) integrable in (0,4].
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...
(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...
(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
Let f : [0,∞) → R be the function defined by f ( x ) = 2 ⌊ x ⌋ − x? where x? = x − ⌊x⌋ is the decimal part of x. Prove that f is injective. Let f: 0,00) + R be the function defined by f(3) = 212) where ã = x — [x] is the decimal part of x. Prove that f is injective.
f:[0,1] -> R |f(x)-f(y)| less than or equal to 4|x-y| Prove f is Riemann-Darboux integrable
(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...
4. Let f be a differentiable function defined on (0, 1) whose derivative is f'(c) = 1 - cos (+) [Note that we can confidently say such an f exists by the FTC.) Prove that f is strictly increasing on (0,1). 5. Let f be defined on [0, 1] by the following formula: 1 x = 1/n (n € N) 0, otherwise (a) Prove that f has an infinite number of discontinuities in [0,1]. (b) Prove that f is nonetheless...
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...