given f is continuous on [0,1]
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Originally Posted by szpengchao given f is continuous on [0,1] this is not true in general. more conditions are needed for this to be true. i believe flyingsquirrel answered a similar question for you earlier. baring that in mind, i don't think it would be hard for you to come up with a counter-example to this
that question is about integrable one. this is a continuous function f!!
Originally Posted by szpengchao that question is about integrable one. this is a continuous function f!! you said continuous on [0,1], that is not enough, as the function defined by is continuous on [0,1] but is not identically zero, yet
should I make it more clear ?
Originally Posted by szpengchao should I make it more clear ? ok, now that makes sense. try a proof by contradiction. assume the integral is zero but f is not. this would mean f is not continuous, a contradiction
assume there exists p in [0,1] , Then we can find a there discontinuous. contradicts.
If is continuous on so is . Suppose that then . What can we say about . Do you see the contradiction in that?
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