Let \( g \) be a bounded function on [0,1] and assume that the restriction of \( g \) to the interval [1/n,1] is Riemann integrable for every \( n \geq 2 \). Show that \( g \) is Riemann integrable over the entire interval [0,1]. (Hint: Let \( \epsilon > 0 \) be given and let \( M > 0 \) be a constant such that \( |g(x)| \leq M \) for all \( x \) in [0,1]. Choose \( n \geq 2 \) such that \( \frac{1}{n} < \frac{\epsilon}{3M} \) and note that $$ \sup\{g(x): x \in [0,1/n]\}1/n - \inf\{g(x): x \in [0,1/n]\}1/n < \frac{2\epsilon}{3} $$ Now use that \( g \) is Riemann integrable on [1/n,1] to find an appropriate partition of [0,1].)
Let \( g \) be a bounded function on [0,1] and assume that the restriction of \( g \) to the interval [1/n,1] is Riemann integrable for every \( n \geq 2 \). Show that \( g \) is Riemann integrable over the entire interval [0,1]. (Hint: Let \( \epsilon > 0 \) be given and let \( M > 0 \) be a constant such that \( |g(x)| \leq M \) for all \( x \) in [0,1]. Choose \( n \geq 2 \) such that \( \frac{1}{n} < \frac{\epsilon}{3M} \) and note that $$ \sup\{g(x): x \in [0,1/n]\}1/n - \inf\{g(x): x \in [0,1/n]\}1/n < \frac{2\epsilon}{3} $$ Now use that \( g \) is Riemann integrable on [1/n,1] to find an appropriate partition of [0,1].)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let \( g \) be a bounded function on [0,1] and assume that the restriction of \( g \) to the interval [1/n,1] is Riemann
(Hint: Let \( \epsilon > 0 \) be given and let \( M > 0 \) be a constant such that \( |g(x)| \leq M \) for all \( x \) in [0,1]. Choose \( n \geq 2 \) such that \( \frac{1}{n} < \frac{\epsilon}{3M} \) and note that
$$ \sup\{g(x): x \in [0,1/n]\}1/n - \inf\{g(x): x \in [0,1/n]\}1/n < \frac{2\epsilon}{3} $$
Now use that \( g \) is Riemann integrable on [1/n,1] to find an appropriate partition of [0,1].)
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