Question

Let $f: [0, 1] \rightarrow \mathbb{R}$ be defined by $$f(x) = \begin{cases} x, & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q}. \end{cases}$$ (a) Calculate the upper and lower Darboux integrals of $f$. (b) Is $f$ integrable? Be sure to justify your answer.

Name: let f 0 1 rightarrow mathbbr be defined by fx begincases x x in mathbbq 0 x notin mathbbq endcases a calculate the upper and lower darboux integrals of f b is f integrable be sure to justify 56606
Uploaded: 2024-12-22T16:51:28-08:00
Duration: 1 min 36 s
Channel: Hoan Nguyen
Description: let f 0 1 rightarrow mathbbr be defined by fx begincases x x in mathbbq 0 x notin mathbbq endcases a calculate the upper and lower darboux integrals of f b is f integrable be sure to justify 56606

          Let $f: [0, 1] \rightarrow \mathbb{R}$ be defined by
$$f(x) = \begin{cases} x, & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q}. \end{cases}$$
(a) Calculate the upper and lower Darboux integrals of $f$.
(b) Is $f$ integrable? Be sure to justify your answer.

let f 0 1 rightarrow mathbbr be defined by fx begincases x x in mathbbq 0 x notin mathbbq endcases a calculate the upper and lower darboux integrals of f b is f integrable be sure to justify 56606

Added by Juan B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Hoan Nguyen

12/22/2024

Step 1

Let $P = \{x_0, x_1, \dots, x_n\}$ be a partition of $[0, 1]$, where $x_0 = 0$ and $x_n = 1$. For each subinterval $[x_{i-1}, x_i]$, we have $$M_i = \sup\{f(x): x \in [x_{i-1}, x_i]\} = \sup\{x: x \in [x_{i-1}, x_i] \cap \mathbb{Q}\} = x_i$$ $$m_i = \inf\{f(x): x Show more…

Show all steps

Thanks for your feedback!

Let $f: [0, 1] \rightarrow \mathbb{R}$ be defined by $$f(x) = \begin{cases} x, & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q}. \end{cases}$$ (a) Calculate the upper and lower Darboux integrals of $f$. (b) Is $f$ integrable? Be sure to justify your answer.