(i) Suppose f : [0, 1]→R and a : [0, 1] →R are continuous functions. Let f(x) >0, for x € [0, 1]and let a be monotonically increasing, differentiable on (0, 1), and that there exists & > 0 such that a'(x) > ɛ for all x E (0, 1). 1 Show that if f(x) da(x) = 0, then f(x) = 0 (ii) Give an example of an a, and a continuous, Riemann-Stieljes integrable function f € R(a) 1 such that [ f(x) da(x) = 0, but f † 0 for some x € [0, 1].
(i) Suppose f : [0, 1]→R and a : [0, 1] →R are continuous functions. Let f(x) >0, for x € [0, 1]and let a be monotonically increasing, differentiable on (0, 1), and that there exists & > 0 such that a'(x) > ɛ for all x E (0, 1). 1 Show that if f(x) da(x) = 0, then f(x) = 0 (ii) Give an example of an a, and a continuous, Riemann-Stieljes integrable function f € R(a) 1 such that [ f(x) da(x) = 0, but f † 0 for some x € [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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![(i) Suppose f : [0, 1]→R and a : [0, 1]–→R are continuous functions.
Let f(x) >0, for x € [0, 1]and let a be monotonically increasing, differentiable on (0, 1), and that
there exists ɛ > 0 such that a'(x) > ɛ for all x E (0, 1).
1
Show that if f(x) da(x) = 0, then f(x) = 0
(ii) Give an example of an a, and a continuous, Riemann-Stieljes integrable function f € R(@)
1
such that [ f(x) da(x) = 0, but f 7 0 for some x E [0, 1].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9444f09-d451-4777-a726-2c28d1bfa98b%2F93bf2873-01c6-44c1-ac96-a96b8169fff4%2Fbmv5uzl_processed.png&w=3840&q=75)
Transcribed Image Text:(i) Suppose f : [0, 1]→R and a : [0, 1]–→R are continuous functions.
Let f(x) >0, for x € [0, 1]and let a be monotonically increasing, differentiable on (0, 1), and that
there exists ɛ > 0 such that a'(x) > ɛ for all x E (0, 1).
1
Show that if f(x) da(x) = 0, then f(x) = 0
(ii) Give an example of an a, and a continuous, Riemann-Stieljes integrable function f € R(@)
1
such that [ f(x) da(x) = 0, but f 7 0 for some x E [0, 1].
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