Thread: do i use integration?

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.

the condition |f(a) -f(b)| is less than
|a-b| for all a, b on [0,1]

Does this not show that 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?

Originally Posted by TD!

Bounded, continuous functions

Perhaps, the use of "bounded" was unnesseray because a countinous function on a closed interval is bounded.

Then I would've needed to include 'on a closed interval'

Originally Posted by TD!

Then I would've needed to include 'on a closed interval'

Good point.


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What about,
boundend and countinous on but not riemann integrable. Thus, you need to include 'on a closed interval' anyway.