if f is bounded on [0,1] and satisfies the condition |f(a) -f(b)| is less than
|a-b| for all a, b on [0,1] , prove that f is Riemann integrable on [0,1]
I have always been slightly confused by Riemann. :confused:
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if f is bounded on [0,1] and satisfies the condition |f(a) -f(b)| is less than
|a-b| for all a, b on [0,1] , prove that f is Riemann integrable on [0,1]
I have always been slightly confused by Riemann. :confused:
Does this not show thatQuote:
the condition |f(a) -f(b)| is less than
|a-b| for all a, b on [0,1]
is countinous of
? Now use the integration theorem which states that every closed countinous function is Riemann integrable.
Yes. it does show f is continuous. But I don't know how to show that it is Riemann integrable. I don't think I ever grasped that when taught.
Can you explain some more?
Bounded, continuous function are Riemann integrable, haven't you seen that?
Perhaps, the use of "bounded" was unnesseray because a countinous function on a closed interval is bounded.Quote:
Originally Posted by TD!
Then I would've needed to include 'on a closed interval' :)
Good point.Quote:
Originally Posted by TD!
--
What about,
boundend and countinous on
but not riemann integrable. Thus, you need to include 'on a closed interval' anyway.