Continuous function with zero integral Let $f$ be a continuous function on $[a,b]$ ($a<b$), such that $\int_{a}^{b}{f(t)dt}=0$
Show that $\exists c\in[a,b], f(c)=0$.

Let $m=\min\{f(x)|x\in[a,b]\}, m=\max\{f(x)|x\in[a,b]\}$(We can get the minimum and maximum because $f$ is continuous on a closed interval) . If $m,M$ have the same sign it can be shown that the integral cant be zero (for example if both are positive, the the integral will be positive). If $m,M$ have different signs apply the intermediate value theorem
