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Problem 1. Prove that a function fis absolutely continuous on [a, b] if and only if it is a conti...
6. The goal of this problem is to prove that a function is Riemann integrable if and only if its ...
exercice 6
6. The goal of this problem is to prove that a function is Riemann integrable if and only if its set of discontinuities has measure 0. So, assume f: a, bR is a bounded function. Define the oscillation of f at , w(f:z) by and for e >0 let Consider the following claims: i- Show that the limit in the definition of the oscillation always exists and that f is continuous at a if and only if w(f;...
If fis an invertible function, which of the following is not true? O If fis continuous,...
If fis an invertible function, which of the following is not true? O If fis continuous, then f - is continuous. O Iff-' is continuous then f is continuous. O If fis decreasing then f-lis decreasing. If f is increasing then f - is decreasing. Differentiate the given function. f(x) = (In x) la 2 O f'(2) — (In z) ha 2 O f'(x) = (n =) (In (In 2) + 1) O f'(x) = { (in (In 2) +...
Real Analysis II Please do it without using Heine-Borel's theorem and do it only if you're sure Problem: Let E be a closed bounded subset of En and r be any function mapping E to (0,∞). Then t...
Real Analysis II
Please do it without using Heine-Borel's theorem
and do it only if you're sure
Problem: Let E be a closed bounded subset of
En and r be any function mapping E to
(0,∞). Then there exists finitely many points yi ∈ E, i
= 1,...,N such that
Here Br(yi)(yi) is the open ball
(neighborhood) of radius r(yi) centered at
yi.
Also, following definitions & theorems should help
that
E CUBy Definition. A subset S of a topological...
real analysis hint 13 Suppose fis a continuous function on R', with period 1. Prove that...
real analysis
hint
13 Suppose fis a continuous function on R', with period 1. Prove that lim Σ f(a)-| f(t) dt 0 for every irrational real number α. Hint: Do it first for f(t)= exp (2nikt), k = 0,±1, ±2, 4.13 Let 2 be the set of functions of form P(t)-Σ_NQC2nikt. The equality holds for functions in . For given ε > 0, there is a P E 2 such that llf-Plloo < ε. Then
solve #5 only please 5 Prove that the function f in problem 4 is integrable and...
solve #5 only please
5 Prove that the function f in problem 4 is integrable and sf = 0. Suggestion: Use the suggestion for problem 4(a) to show that given €>0, there is a partition Pof [0, 1] with Uff, P) < 2€ , while Laf, P) =0. Do this by enclosing the points of the finite set where f(x) 2e in a finite set of disjoint closed intervals, each contained in (0,1), with the sum of the lengths <€....
Problem 1. Suppose that f:(a,b) + R is a continuous function and there exists a point...
Problem 1. Suppose that f:(a,b) + R is a continuous function and there exists a point p e (a, b) such that f' exists and is bounded on (a,b) {p}. Prove that f is uniformly continuous on (a,b).
Problem 3. Read about compactness in Section 2.8 of the book. Then, prove, WITHOUT RELYING ON HEINE-BOREL's THEOREM, the following. Let E be a closed bounded subset of E and r be any function...
Problem 3. Read about compactness in Section 2.8 of the book. Then, prove, WITHOUT RELYING ON HEINE-BOREL's THEOREM, the following. Let E be a closed bounded subset of E and r be any function mapping E to (0,00). Then there ensts finitely many pints yi E E,i = 1, , N such that i-1 Here Br(y.)(y) is the open ball (neighborhood) of Tudius r(y.) centered at yi.
Problem 3. Read about compactness in Section 2.8 of the book. Then, prove,...
1. Let f:R → R be the function defined as: 32 0 if x is rational...
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous...
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous on a compact set D. Then f is uniformly continuous on D. (a) Suppose that f is not uniformly continuous on D. Then there exists an for every n EN there exists xn and > 0 such that yn in D with la ,-ynl < 1/n and If(xn)-f(yn)12 E. (b) Apply 4.4.7, every bounded sequence has a convergent subsequence, to obtain a convergent subsequence...
(12) A function f: A R is called a step function if ran(f) is finite. Prove...
(12) A function f: A R is called a step function if ran(f) is finite. Prove that for every R, there is a sequence fn: [a, b-R of step functions such continuous function f [a, b that fn(x)f(x) for all e la,b and fnfuniformly [a, b). on
(12) A function f: A R is called a step function if ran(f) is finite. Prove that for every R, there is a sequence fn: [a, b-R of step functions such continuous function...
exercice 6
6. The goal of this problem is to prove that a function is Riemann integrable if and only if its set of discontinuities has measure 0. So, assume f: a, bR is a bounded function. Define the oscillation of f at , w(f:z) by and for e >0 let Consider the following claims: i- Show that the limit in the definition of the oscillation always exists and that f is continuous at a if and only if w(f;...
If fis an invertible function, which of the following is not true? O If fis continuous, then f - is continuous. O Iff-' is continuous then f is continuous. O If fis decreasing then f-lis decreasing. If f is increasing then f - is decreasing. Differentiate the given function. f(x) = (In x) la 2 O f'(2) — (In z) ha 2 O f'(x) = (n =) (In (In 2) + 1) O f'(x) = { (in (In 2) +...
Real Analysis II
Please do it without using Heine-Borel's theorem
and do it only if you're sure
Problem: Let E be a closed bounded subset of
En and r be any function mapping E to
(0,∞). Then there exists finitely many points yi ∈ E, i
= 1,...,N such that
Here Br(yi)(yi) is the open ball
(neighborhood) of radius r(yi) centered at
yi.
Also, following definitions & theorems should help
that
E CUBy Definition. A subset S of a topological...
real analysis
hint
13 Suppose fis a continuous function on R', with period 1. Prove that lim Σ f(a)-| f(t) dt 0 for every irrational real number α. Hint: Do it first for f(t)= exp (2nikt), k = 0,±1, ±2, 4.13 Let 2 be the set of functions of form P(t)-Σ_NQC2nikt. The equality holds for functions in . For given ε > 0, there is a P E 2 such that llf-Plloo < ε. Then
solve #5 only please
5 Prove that the function f in problem 4 is integrable and sf = 0. Suggestion: Use the suggestion for problem 4(a) to show that given €>0, there is a partition Pof [0, 1] with Uff, P) < 2€ , while Laf, P) =0. Do this by enclosing the points of the finite set where f(x) 2e in a finite set of disjoint closed intervals, each contained in (0,1), with the sum of the lengths <€....
Problem 1. Suppose that f:(a,b) + R is a continuous function and there exists a point p e (a, b) such that f' exists and is bounded on (a,b) {p}. Prove that f is uniformly continuous on (a,b).
Problem 3. Read about compactness in Section 2.8 of the book. Then, prove, WITHOUT RELYING ON HEINE-BOREL's THEOREM, the following. Let E be a closed bounded subset of E and r be any function mapping E to (0,00). Then there ensts finitely many pints yi E E,i = 1, , N such that i-1 Here Br(y.)(y) is the open ball (neighborhood) of Tudius r(y.) centered at yi.
Problem 3. Read about compactness in Section 2.8 of the book. Then, prove,...
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous on a compact set D. Then f is uniformly continuous on D. (a) Suppose that f is not uniformly continuous on D. Then there exists an for every n EN there exists xn and > 0 such that yn in D with la ,-ynl < 1/n and If(xn)-f(yn)12 E. (b) Apply 4.4.7, every bounded sequence has a convergent subsequence, to obtain a convergent subsequence...
(12) A function f: A R is called a step function if ran(f) is finite. Prove that for every R, there is a sequence fn: [a, b-R of step functions such continuous function f [a, b that fn(x)f(x) for all e la,b and fnfuniformly [a, b). on
(12) A function f: A R is called a step function if ran(f) is finite. Prove that for every R, there is a sequence fn: [a, b-R of step functions such continuous function...
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