continuous function on a closed and bounded set

**Kat-M** · August 10th 2009, 02:21 PM

Let $f$ be 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 $[a,b]$ .

**siclar** · August 10th 2009, 02:58 PM

Originally Posted by Kat-M

Let $f$ be 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 $f(x)=c\neq 0$ for 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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