Question

4. Let $f$ be the function defined by $f(x) = \begin{cases} x^2 + 6 & \text{if } x \text{ is rational} \\ -5x & \text{if } x \text{ is irrational} \end{cases}$ (a) At what values of $x$ is $f$ continuous? (b) Over the interval $[-1, 1]$, show that $f$ is Lebesgue integrable, but not Riemann integrable.

          4. Let $f$ be the function defined by
$f(x) = \begin{cases} x^2 + 6 & \text{if } x \text{ is rational} \\ -5x & \text{if } x \text{ is irrational} \end{cases}$
(a) At what values of $x$ is $f$ continuous?
(b) Over the interval $[-1, 1]$, show that $f$ is Lebesgue integrable, but not Riemann integrable.

4. Let f be the function defined by
f(x) = x^2 + 6 if x is rational
-5x if x is irrational
(a) At what values of x is f continuous?
(b) Over the interval [-1, 1], show that f is Lebesgue integrable, but not Riemann integrable.

Added by Christine J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

07/05/2023

Step 1

It is defined as $x^2 + 6$ when $x$ is rational and $-5x$ when $x$ is irrational. Show more…

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4. Let $f$ be the function defined by $$f(x) = \begin{cases} x^2 + 6 & \text{if } x \text{ is rational} \\ -5x & \text{if } x \text{ is irrational}. \end{cases}$$ (a) At what values of $x$ is $f$ continuous? (b) Over the interval $[-1,1]$, show that $f$ is Lebesgue integrable, but not Riemann integrable.