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 $\exists c\in[a,b], f(c)=0$.

By the Fundamental Theorem of Calculus, the function
$$
F(x):=\int_a^xf(t)dt
$$
is continuous on $[a,b]$ with derivative $F'=f$ on $(a,b)$.
Now $F(a)=F(b)=0$.
By Rolle, there exists $c\in (a,b)$ such that $F'(c)=f(c)=0$.
