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E. Define f(x) = %* sink dt. (a) Prove that this integral is defined. (6) Compute...
6. Let f be a continuous function on R and define F(z) = | r-1 f(t)dt...
6. Let f be a continuous function on R and define F(z) = | r-1 f(t)dt x E R. Show that F is differentiable on R and compute F'
real analysis II. Consider the function f:[0,1] - R defined by f(x) 0 if x E...
real analysis
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q.
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
6) If E is any countable subset of real numbers prove that A*(E) = A*(E) = 0. 7) Show that the se...
6) If E is any countable subset of real numbers prove that A*(E) = A*(E) = 0. 7) Show that the set of all real numbers IR is measurable with >(IR) = . 8) Prove that If f : [a, b] IR is continuous [a; b]then it is measurable [a, b]. 9) Give an example of a function f : [O, 1] IR which is measurable on [O, 1] but not continuos on [O, 1]. 10) Find the Lebesgue integral...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f...
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to f...
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...
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous, Let X and Y be top...
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
6 that is continuous on the entire Note that F(x) = J-6 V44 +6 dt. So...
6 that is continuous on the entire Note that F(x) = J-6 V44 +6 dt. So assume f(e) = V 24 + 6 real line. Use the Second Fundamental Theorem of Calculus, which states that, if f is continuous on an open interval I containing a, then for every x in the interval, d f(x). f(t) dt = dx Step 2 In this problem, F(x) = Live* +6 dt. Therefore, F'(x) = f(t) dt = f(x) = V Submit Skip...
let f:IR-DIR be defined by f(x) = 31x), Prove that f(x)-31x1 is invertible. Compute the preimage...
let f:IR-DIR be defined by f(x) = 31x), Prove that f(x)-31x1 is invertible. Compute the preimage f'((2,4]).
6. Let f be a continuous function on R and define F(z) = | r-1 f(t)dt x E R. Show that F is differentiable on R and compute F'
real analysis
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q.
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
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...
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
6 that is continuous on the entire Note that F(x) = J-6 V44 +6 dt. So assume f(e) = V 24 + 6 real line. Use the Second Fundamental Theorem of Calculus, which states that, if f is continuous on an open interval I containing a, then for every x in the interval, d f(x). f(t) dt = dx Step 2 In this problem, F(x) = Live* +6 dt. Therefore, F'(x) = f(t) dt = f(x) = V Submit Skip...
let f:IR-DIR be defined by f(x) = 31x), Prove that f(x)-31x1 is invertible. Compute the preimage f'((2,4]).
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