Question

Show that Thomae's function is Darboux/Riemann integrable and its integral is equal to 0.

Answer #1

For the given function f(x) = c show that it is
Riemann integrable on the interval [0, 1] and find the
Riemann integral

Show that a bounded function which has at most a finite number
of discontinuities is Riemann integrable

Real Analysis:
Give an example of a function that is not Riemann Integral.

Suppose that ff is a Riemann integrable function on [0,2][0,2]
and that ∫20f(x)dx=5∫02f(x)dx=5. Suppose further that AA is a
function such that if 0≤a≤20≤a≤2 then the average value of ff on
the interval [0,a][0,a] is given by A(a)A(a). Find a formula for
the average value of ff on [a,2][a,2] using AA.

Give an example of a bounded unsigned function on [0,1] that is
Lebesgue integrable but not Riemann integrable. Briefly justify why
those properties hold, using theorems and definitions from the
textbook.

Consider the integral
∫12 0 (2?^2+3?+2)??
(a) Find the Riemann sum for this integral using left endpoints
and ?=4
L4=
(b) Find the Riemann sum for this same integral, using right
endpoints and ?=4
R4=

Let f be a monotonic increasing function on a closed interval
[a, b]. Show that f is
Riemann integrable on [a, b].

Let f: [0, 1] --> R be defined by f(x) := x. Show that f is
in Riemann integration interval [0, 1] and compute the integral
from 0 to 1 of the function f using both the definition of the
integral and Riemann (Darboux) sums.

Express the given integral as the limit of a Riemann sum but do
not evaluate: the integral from 0 to 3 of the quantity x cubed
minus 6 times x, dx.

Assumethatf ∈C[a,b]isacontinuousfunctionandP ={a=x0 <x1
<···<xn =b}isapartitionof[a,b].
Write precise definitions of the following.
(a) The the upper sum U(f,P) and the Lower sum L(f,P) are
defined as .........[5 points]
(b) The lower Riemann Integral L(f) and the Upper Riemann Integral
U(f) are defined as ..... [5 points]
(c) The function f : [a, b] ?→ R is called Riemann Integrable
when ........ [5 points]

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