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In this problem, we will look at some properties of the Riemann integral of the absolute value of a function. (a) Suppose f€R[a, b]. Show that |f € R[a, b], and that -b 0≤||²₁|≤ [²₁₁ |f| a (b) Find an example of a function ƒ : [a, b] → R such that |ƒ| € R[a, b] but ƒ ‡ R[a, b]. (Hint: Think about the Dirichlet function) (c) Suppose f: [a, b] → R is continuous. Show that if f(c) > 0 for some c € [a, b], then there exists some > 0 such that rc+d T if and only if f(x) = 0 for all x = [a, b] (Remark: Try to be careful about strict/non-strict inequalities and open/closed inter- vals in this part.) (d) Suppose f: [a, b] → R is continuous. Show that L₁²₁1₁=0 ƒ> 0 a

In this problem, we will look at some properties of the Riemann integral of the absolute value of a function. (a) Suppose f€R[a, b]. Show that |f € R[a, b], and that -b 0≤||²₁|≤ [²₁₁ |f| a (b) Find an example of a function ƒ : [a, b] → R such that |ƒ| € R[a, b] but ƒ ‡ R[a, b]. (Hint: Think about the Dirichlet function) (c) Suppose f: [a, b] → R is continuous. Show that if f(c) > 0 for some c € [a, b], then there exists some > 0 such that rc+d T if and only if f(x) = 0 for all x = [a, b] (Remark: Try to be careful about strict/non-strict inequalities and open/closed inter- vals in this part.) (d) Suppose f: [a, b] → R is continuous. Show that L₁²₁1₁=0 ƒ> 0 a

Advanced Engineering Mathematics
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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analysis please answer a-d

) In this problem, we will look at some properties of the Riemann integral of the absolute value of a function. (a) Suppose fER[a, b]. Show that If € R[a, b], and that 0≤ a (b) Find an example of a function f : [a, b] → R such that f € R[a, b] but f & R[a, b]. (Hint: Think about the Dirichlet function) a : (c) Suppose f [a, b] → R is continuous. Show that if f(c) > 0 for some c € [a, b], then there exists some > 0 such that Let's c-8 if and only if f(x) = 0 for all x = [a, b] f> 0 (Remark: Try to be careful about strict/non-strict inequalities and open/closed inter- vals in this part.) (d) Suppose f: [a, b] → R is continuous. Show that |f|= = 0
Transcribed Image Text:) In this problem, we will look at some properties of the Riemann integral of the absolute value of a function. (a) Suppose fER[a, b]. Show that If € R[a, b], and that 0≤ a (b) Find an example of a function f : [a, b] → R such that f € R[a, b] but f & R[a, b]. (Hint: Think about the Dirichlet function) a : (c) Suppose f [a, b] → R is continuous. Show that if f(c) > 0 for some c € [a, b], then there exists some > 0 such that Let's c-8 if and only if f(x) = 0 for all x = [a, b] f> 0 (Remark: Try to be careful about strict/non-strict inequalities and open/closed inter- vals in this part.) (d) Suppose f: [a, b] → R is continuous. Show that |f|= = 0
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