# Thread: continuous function on a closed and bounded set

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

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 $[a,b]$.

2. 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?