For a function h and an interval I, the oscillation of h on I is defined by w(h, I) = sup |h(æ) – h(y)|- x,YEI 1. For any two bounded functions, show that w(fg, I) < sup |f| -w(g, I) + sup |g| - w(f,I). I I
For a function h and an interval I, the oscillation of h on I is defined by w(h, I) = sup |h(æ) – h(y)|- x,YEI 1. For any two bounded functions, show that w(fg, I) < sup |f| -w(g, I) + sup |g| - w(f,I). I I
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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(Second picture shows a definition)
![Properlins uf the integral.
Recalf-
Thm= Let f be bouneleel. Then fE R [a,b]f end ouly if
V E>0, ] purtolion P st.l ulf P)- L(f, P)<s
Lit Pia= Xo< X, <
< Xn =b. Then Ucf. P)- L f.P)
|
Z Mi ( X; - i-)
ン」
sup f.
讨 f
inf
Miz
m; =
in
ī (Mi- Mi) oX;
wilf) o Xj,
wilf)= _sup {- inf f
ニ
is called the oscillation uff over
[xi-, Xi].
So the intifruhtily creterion cam also be statiel as:
f is bondeel. Then f E RIa,b]
iff Hs, J P st.
Ž wilf) ox; c {
If f orillates
too much, then fis not integrable.
One esxample:
dix)=
x+Q.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8c2ad84-3927-490a-bf66-2d9eb7f56b39%2Fcfd37ab5-9880-42b3-8c39-4943ea2307fd%2Fauq5j6b_processed.png&w=3840&q=75)
Transcribed Image Text:Properlins uf the integral.
Recalf-
Thm= Let f be bouneleel. Then fE R [a,b]f end ouly if
V E>0, ] purtolion P st.l ulf P)- L(f, P)<s
Lit Pia= Xo< X, <
< Xn =b. Then Ucf. P)- L f.P)
|
Z Mi ( X; - i-)
ン」
sup f.
讨 f
inf
Miz
m; =
in
ī (Mi- Mi) oX;
wilf) o Xj,
wilf)= _sup {- inf f
ニ
is called the oscillation uff over
[xi-, Xi].
So the intifruhtily creterion cam also be statiel as:
f is bondeel. Then f E RIa,b]
iff Hs, J P st.
Ž wilf) ox; c {
If f orillates
too much, then fis not integrable.
One esxample:
dix)=
x+Q.
![For a function h and an interval I, the oscillation of h on I is defined by
w(h, I) = sup |h(æ) – h(y)|-
x,YEI
1. For any two bounded functions, show that
w(fg,I) < sup |f| -w(g, I) + sup |g| - w(f,I).
I
I
2. Let f, g € R[a, b]. Show that f - g€ R[a,b].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8c2ad84-3927-490a-bf66-2d9eb7f56b39%2Fcfd37ab5-9880-42b3-8c39-4943ea2307fd%2Fvexh6a1_processed.png&w=3840&q=75)
Transcribed Image Text:For a function h and an interval I, the oscillation of h on I is defined by
w(h, I) = sup |h(æ) – h(y)|-
x,YEI
1. For any two bounded functions, show that
w(fg,I) < sup |f| -w(g, I) + sup |g| - w(f,I).
I
I
2. Let f, g € R[a, b]. Show that f - g€ R[a,b].
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