![(c) [5 points] Prove that f(r) [5 p ) = Σ (-1-rn oints Prove that f(x converges uniformly on [-c, c when 0<c<1. lenny](http://img.homeworklib.com/questions/4bab25f0-0e9d-11ec-8fc6-57f35267e1e1.png?x-oss-process=image/resize,w_560)
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A function with domain which is a interval of real number can not be uniformly convergent but can be uniformly continuous... And the given function is uniformly continuous... And we can prove that...
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(c) [5 points] Prove that f(r) [5 p ) = Σ (-1-rn oints Prove that f(x...
The series Σ 1 np converges if and only if p < 1 Select one: O...
The series Σ 1 np converges if and only if p < 1 Select one: O True O False
5. (10 points) Let p="x < y", q="x < 1", and r="y > 0". Using ~,...
5. (10 points) Let p="x < y", q="x < 1", and r="y > 0". Using ~, 1, V write the following statements in terms of the symbols p, q, and r. (a) 0 <y < x < 1. (b) 1 < x <y<0.
5. Let f,lr)- x *a. Show that {h} converges uniformly to 0 on [0, a] for...
5. Let f,lr)- x *a. Show that {h} converges uniformly to 0 on [0, a] for any a, 0 < a < 1. b. Does {f,) converge uniformly on [0, 1]?
5. Let be the function defined by f(x) = -1 3 1.5 if r <0 if...
5. Let be the function defined by f(x) = -1 3 1.5 if r <0 if 0<x<2 if 3 < r <5 Find the Lebesgue integral of f over (-10,10).
(5) Let qe Q. Suppose that a <b, 0<c<d, and that f : [a, b] →...
(5) Let qe Q. Suppose that a <b, 0<c<d, and that f : [a, b] → [c, d]. If f is integrable on [a,b], then prove that * (t)dt) = f'(x) for all 3 € (a, b).
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is...
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
(h) Define f : [0, 2] + R by 122 if 0 <<<1 f(x) = {...
(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.
1 pt) A P(X1126) Probability B. P(X < 966) Probability c. P(X > 1046) Probability
1 pt) A P(X1126) Probability B. P(X < 966) Probability c. P(X > 1046) Probability
5. Is f continuous at f(1)? (10 points) [-x2 +1, 4x, f(x) = -5, -1<x<0 0<x<1...
5. Is f continuous at f(1)? (10 points) [-x2 +1, 4x, f(x) = -5, -1<x<0 0<x<1 x=1 1<x<3 3<x<5 - 4x + 8 1,
1. Let x, a € R. Prove that if a <a, then -a < x <a.
1. Let x, a € R. Prove that if a <a, then -a < x <a.
The series Σ 1 np converges if and only if p < 1 Select one: O True O False
5. (10 points) Let p="x < y", q="x < 1", and r="y > 0". Using ~, 1, V write the following statements in terms of the symbols p, q, and r. (a) 0 <y < x < 1. (b) 1 < x <y<0.
5. Let f,lr)- x *a. Show that {h} converges uniformly to 0 on [0, a] for any a, 0 < a < 1. b. Does {f,) converge uniformly on [0, 1]?
5. Let be the function defined by f(x) = -1 3 1.5 if r <0 if 0<x<2 if 3 < r <5 Find the Lebesgue integral of f over (-10,10).
(5) Let qe Q. Suppose that a <b, 0<c<d, and that f : [a, b] → [c, d]. If f is integrable on [a,b], then prove that * (t)dt) = f'(x) for all 3 € (a, b).
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
(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.
1 pt) A P(X1126) Probability B. P(X < 966) Probability c. P(X > 1046) Probability
5. Is f continuous at f(1)? (10 points) [-x2 +1, 4x, f(x) = -5, -1<x<0 0<x<1 x=1 1<x<3 3<x<5 - 4x + 8 1,
1. Let x, a € R. Prove that if a <a, then -a < x <a.
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