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Math Help - continuous function on a closed and bounded set

  1. #1
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    continuous function on a closed and bounded set

    Let fbe continuous on [a,b]. If f(x)\leq 0 on [a,b] and \int_a^b f(x)dx=0, then show that f(x)\equiv 0 on [a,b].
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  2. #2
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    Quote Originally Posted by Kat-M View Post
    Let fbe continuous on [a,b]. If f(x)\leq 0 on [a,b] and \int_a^b f(x)dx=0, then show that f(x)\equiv 0 on [a,b].
    Use continuity! I.E. Suppose f(x)=c\neq 0for some x\in [a,b]. Then by continuity, there exists a small interval about x in which the value of f on this interval varies by less than, say, c/2. What can we say about the integral over this interval?
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