Prove that if the integral of a function is zero, then it is zero for at least one point

by RoRi

September 10, 2015

Let $f$ be a continuous function on an interval $[a,b]$ . (Assume $b > a$ .) Then, if

$\int_a^b f(x) \, dx = 0,$

prove that $f(c) = 0$ for at least one $c \in [a,b]$ .

Proof. Since $f$ is continuous on the interval $[a,b]$ we may apply the mean value theorem (Theorem 3.15 in Apostol) to conclude

$\int_a^b f(x) \, dx = f(c) (b-a) \qquad \text{for some } c \in [a,b].$

So,

$\int_a^b f(x) \, dx = 0 \quad \implies \quad f(c) (b-a) = 0.$

But since $b > a$ , we know $(b-a) \neq 0$ ; hence, we must have $f(c) = 0$ for some $c \in [a,b]. \qquad \blacksquare$

Stumbling Robot