The unit step function IR→ R is defined as if x < 0 { if x > 0. I(x) = 0 1 (i) Let s € (a, b), and suppose f : [a, b] → R is bounded on [a, b] and continu- ous at s. Let a(x) = I(x-s). Prove that f is Riemann-Stieltjes integrable with respect to a, and compute its integral.
The unit step function IR→ R is defined as if x < 0 { if x > 0. I(x) = 0 1 (i) Let s € (a, b), and suppose f : [a, b] → R is bounded on [a, b] and continu- ous at s. Let a(x) = I(x-s). Prove that f is Riemann-Stieltjes integrable with respect to a, and compute its integral.
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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![The unit step function \( I : \mathbb{R} \to \mathbb{R} \) is defined as
\[
I(x) =
\begin{cases}
0 & \text{if } x \leq 0 \\
1 & \text{if } x > 0
\end{cases}
\]
(i) Let \( s \in (a, b) \), and suppose \( f : [a, b] \to \mathbb{R} \) is bounded on \([a, b]\) and continuous at \( s \). Let \( \alpha(x) = I(x-s) \). Prove that \( f \) is Riemann-Stieltjes integrable with respect to \(\alpha\), and compute its integral.
(ii) Let \( s_1, s_2, \ldots, s_n \) be distinct points in \((a, b)\), and suppose that \( f \) is continuous on \([a, b]\). Compute the Riemann-Stieltjes integral of \( f \) with respect to \(\alpha : \mathbb{R} \to \mathbb{R}\) defined as \(\alpha(x) = \sum_{i=1}^{n} I(x - s_i)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fc3e8d5cd-5b0d-4557-afae-80232863885b%2F2odttqr_processed.png&w=3840&q=75)
Transcribed Image Text:The unit step function \( I : \mathbb{R} \to \mathbb{R} \) is defined as
\[
I(x) =
\begin{cases}
0 & \text{if } x \leq 0 \\
1 & \text{if } x > 0
\end{cases}
\]
(i) Let \( s \in (a, b) \), and suppose \( f : [a, b] \to \mathbb{R} \) is bounded on \([a, b]\) and continuous at \( s \). Let \( \alpha(x) = I(x-s) \). Prove that \( f \) is Riemann-Stieltjes integrable with respect to \(\alpha\), and compute its integral.
(ii) Let \( s_1, s_2, \ldots, s_n \) be distinct points in \((a, b)\), and suppose that \( f \) is continuous on \([a, b]\). Compute the Riemann-Stieltjes integral of \( f \) with respect to \(\alpha : \mathbb{R} \to \mathbb{R}\) defined as \(\alpha(x) = \sum_{i=1}^{n} I(x - s_i)\).
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