Answer to Question #200746 in Real Analysis for Rabia qadeer

Question

Define a partition of an interval. Write any three different examples of

partitions of [0,1].

(ii). Let a real valued function 𝑓 be defined and bounded over [𝑎, 𝑏]. Prove that 𝑓 is

Riemann integrable over [𝑎, 𝑏] if for each 𝜖 > 0 there is a partition 𝑃 such that

𝑈(𝑃, 𝑓) − 𝐿(𝑃, 𝑓) < 𝜖

(iii). Show that 𝑓(𝑥) = 𝑥2

is Riemann integrable over [0,2].

Answer 1

Ans:-

$(i)$ Partition of a Set is defined as "A collection of disjoint subsets of a given set. The union of the subsets must equal the entire original set".

three different examples of partitions of $[0,1]$ are $(0,0.4) \ , \ (0.4,0.6)\ , \ (0.6,1)$

$(ii)$ Let f  be bounded on $[a,b]$ . Then f  is Riemann integrable if and only if for every  ϵ there is a partition on $[a,b]$  such that: $0≤U(f,P)−L(f,P)≤ϵ$

Since

$\int_ a ^b f=inf[{U(f,P)}]=sup[{L(f,P)}]$

there should exist two partitions $P_1 \ and \ P_2$  such that:

$0≤U(f,P_1)−\int _a ^b f<\dfrac{ϵ}2$

and

$0≤\int^b_af−L(f,P_2)<\dfracϵ2$

The argument is almost identical for $L(f,P)$ (except flipping some inequalities)

Hence the function $f$ is Riemann integrable

$(iii) \ f(x)= x^2$

For Reimann integrable

$\Rightarrow ∫^b _a f(x)dx$

So We have this expression equal to

$\Rightarrow ∫^b _0 x^2dx−∫^a _0 x^2dx$

$\Rightarrow (b^3−a^3)∫^1_0 x^2dx$

Here $a=0 \ \ and \ \ b=2$

Hence $𝑓(𝑥) = 𝑥^2$ is Riemann integrable over $[0,2]$