Real Analysis :Problem Prove a function is integrable over [a,b]

Let f:[a,b] mapped to the Reals be a function that is integrable over [a,b], and let g:[a,b] mapped to the Reals be a function that agrees with f except at two points. Prove g is integrable over [a,b].

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Let the function h(x)= f(x)-g(x) on [a,b].
Then the function h(x) is exactly 0 except for the two ...

Solution Summary

A function is proven to be integrable over an interval.

We have just finished up integration and are done with a first course in analysis, so chapters 1-6 of Rudin. We are also using the Ross and Morrey/Protter book. Please answer question fully and clearly explaining every step. Any solution short of perfect is useless to me. So if you are not 100% sure whether your answer is right,

Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and continuously differentiable on [a,b]. Let {p be element of P} be a partition of [a,b], and define
U(f,g,p) = SIGMA (Mi(g(the ith term of x) - g(the (i-1)th term of x))) as i=1 to n
L(f,g,p) = SIGMA (Ni(g(the ith term of x)-g(the (i-1)th term of x))) as i=1 t

Prove theorem 7.3 in notes attached.
Section 7: The Lebesgue Integral
Definition 7.1 Let L be the set of real-valued functions f such that for some g and h in f=g-h almost everywhere. The set L is called the set of Lebesgue integrablefunction on and the Lebesgue integral of f is defined as follows: .
Theorem 7

Suppose the continuous function f:[a,b]->R has the property that:
The integral from c to d f<=0 whenever a<=cProve that f(x)<=0 for all x in [a,b]. Is this true if we require only integrability of the function?

Please see the attached file for the fully formatted problems.
Prove that if f is integrable on [0, 1], then
lim n !1 Z
1
0
x n f(x)dx = 0
Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrability on R , where they define partition, refinement of a partition,

Q. Show that f is Riemann-integrable. What is ∫[0,1] f(x)dx?
(Hint: What's the set of discontinuity of f? Does it have Vol1-zero?)
Please see attached for full question.

Suppose f is Reimann integrable on [a,b] and let F(x)= ∫_a^x▒f(t)dt for all x ∈[a,b]. Prove that F is continuous on [a,b] (hint: f must be bounded)
Let F(x) = {█(x^2 sin(1/x) if 0<|x|≤1,@0 if x=0)┤ and let f(x) = F'(x)
Prove that F'(x) exists for all x ∈[-1,1]
Find f(x) for all x ∈[-1,1], and prove t

Please see the attached file for the fully formatted problems.
1.
? Calculate the Taylor Polynomial and the Taylor residual for the function .
? Prove that as , for all .
? Find the Taylor series of f.
? What is the radius of convergence for the Taylor series? Justify your answer.
2.
? Let f:[0,1] be a bo

We have learned Rolle, Lagrange, Fermat, Taylor Theorems in our RealAnalysis class and we have finished differentiation. We just started integration. In this problem we are not supposed to use any material we haven't learned, ie integration. We are using the books by Rudin, Ross, Morrey/Protter.
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