Question involving integration and irrationality

Question

I have a little question on integral and irrationality :

Imagine you have: $$\int_{a}^{b}f(x)dx=p$$

Where $p$ is irrational

Can we deduce that there is $b\leq \varepsilon \leq a$ such that $f(\varepsilon)$ is irrational ?

Thanks a lot for your time

Answer 1

No. Let $a=0, b=\sqrt 2, f(x)=1$.

Answer 2

Hint: take $b - a$ irrational, $f =$ some constant...