Prove the statement for definite integral We have positive continuous function $f(x)$ defined on $\mathbb{R}$,
such as $\int_{-\infty}^{+\infty} f(x) dx = 1$
let $\alpha \in (0,1)$ and $[a,b]$ is an interval
 of minimal length amongst intervals for those holds: $\int_{a}^{b} f(x) dx = \alpha$.
Task is to prove that $f(a) = f(b)$.  
I managed to sum up following statements:
$F(\infty) - F(-\infty) = 1$
$F(b) - F(a) = \alpha$
And one to prove is: $f(b) - f(a) = 0$
I am looking for hints to build a proof here. 

If $f(a)\neq f(b)$, say $f(a)$ is larger. Then we can always shift the interval towards the direction of $a$ to make it shorter (since $f$ is continuous).
