Question

1 Let $\chi_Q(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \\ 0 & \text{if } x \notin \mathbb{Q} \end{cases}$ Prove that $\chi_Q$ is not Riemann integrable on $[0, 1]$. (Hint: choose your sample points intelligently. Intuitively, the Riemann integral fails here because it cannot accommodate this many discontinuities. This issue will be further analyzed in MATH 425).

          1 Let
$\chi_Q(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \\ 0 & \text{if } x \notin \mathbb{Q} \end{cases}$
Prove that $\chi_Q$ is not Riemann integrable on $[0, 1]$. (Hint: choose
your sample points intelligently. Intuitively, the Riemann integral
fails here because it cannot accommodate this many discontinuities.
This issue will be further analyzed in MATH 425).

1 Let
(x) = 1 if x ∈ℚ
0 if x ∉ℚ
Prove that is not Riemann integrable on [0, 1]. (Hint: choose
your sample points intelligently. Intuitively, the Riemann integral
fails here because it cannot accommodate this many discontinuities.
This issue will be further analyzed in MATH 425).

Added by Felix B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Shaiju T

06/02/2022

Step 1

Step 1: For the function \chi_{Q}(x), we can choose the partition P = {0, 1/2, 1}. Show more…

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1. Let chi_{Q}(x) = egin{cases} 1 & ext{if } x in Q, \ 0 & ext{if } x otin Q. end{cases} Prove that chi_{Q} is not Riemann integrable on [0,1]. (Hint: choose your sample points intelligently. Intuitively, the Riemann integral fails here because it cannot accommodate this many discontinuities. This issue will be further analyzed in MATH 425). 2. Let f(x) = egin{cases} 1 & ext{if } x in mathbb{Q}, \ 0 & ext{if } x otin mathbb{Q}. end{cases} Prove that f(x) is not Riemann integrable on [0,1]. (Hint: choose your sample points intelligently. Intuitively, the Riemann integral fails here because it cannot accommodate this many discontinuities. This issue will be further analyzed in MATH 425).