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 ...

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A function is proven to be integrable over an interval.

If f is a reimann integrablefunction on [a,b], and if [c,d] is a subset of [a,b], prove that f is reimann integrable on [c,d]
hint: if P is any partition of [c,d], P can be extended to a partition P* of [a,b] with ||P*|| <= ||P||. Show that
U(f,P) - L(f,P) <= U(f,P*) - L(f,P*)

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 be a realfunction on [a, b]. Suppose that f is Riemann integrable on
[c, b] for every a < c < b.
(a) Show that if f is also Riemann-integrable on
[a, b] then integral b-a(f dx) = limc-a integral b-c(f dx).
(b) Give an example of a function g on [a, b] for which limc-a integral b-c(g dx) is defined, while g is n

Are these functions Reimann Integrable? I am just learning this topic, so my description may not be accurate. A function is Reimann Integrable if it's Upper Darboux Sums and Lower Darboux suns are equal.
Or stated another way, if U(f, P) - L(f, P) < e
The two functions are piecewise functions.
1) f(x) = { 0 when x =

Suppose that the function f:[a,b]->R is integrable and there is a postive number m such that f(x) >= m for all x in [a,b]. Show that the reciprocal function 1/f:[a,b]->R is integrable by proving that for each partition P of the interval [a,b],
U(1/f,P) - L(1/f,P) <= 1/m^2[U(f,P) - L(f,P)]

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

We have just finished up integration and are done with a first course in analysis, 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,

1. Let f: [a,b] ----> R and suppose f is integrable with respect to alpha. Prove that for any c in the real numbers, cf is integrable with respect to alpha and the integral from a to b of cf d(alpha) = c times integral from a to b of f d(alpha).
Give an example of a function f : [0,1] ----> R such that f is Riemann integra

For what values of a in R (real numbers) is the function (1+x^2)^a in L^2
NOTE: let L^2 be like in Lebesgue integration where the set of all measurable functions that are square-integrable forms a Hilbert space, the so-called L2 space.
keywords: integration, integrates, integrals, integrating, double, triple, multiple, r