If is a continuous function on such that for all continuous functions . Then on .

So yeah, I know I have to invoke that but after that I'm stumped. Thanks.

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- May 14th 2010, 04:03 PMPinkkProving a function is identically zero
If is a continuous function on such that for all continuous functions . Then on .

So yeah, I know I have to invoke that but after that I'm stumped. Thanks. - May 14th 2010, 04:09 PMmabruka
Here is an idea.

We have where , so on .

Hence f=0 on [a,b]

What do you think? - May 14th 2010, 04:10 PMPinkk
I think that makes sense, yeah.

- May 14th 2010, 04:43 PMPinkk
And I just want to make sure that proving there cannot be some point where . So suppose there is (at least) one point such that . Then since is continuous on , on some interval , so on the closed interval . Clearly since the function must achieve a minimum at a point on the interval by its continuity on a bounded interval, so for any partition , , so and since the function is zero everywhere else, then , which is a contradiction.

I think this seems a little overboard but I just want to make sure if this is correct or if there is an easier proof that shows cannot be positive at any point. Thanks. - May 14th 2010, 04:48 PMDrexel28