If for every sub-segment of $[a,b]$ exists $x$ so that $f(x)\le1$ prove that integral in that segment < $a-b$

Question

I've been trying to solve this question:

Let $f(x)$ be an integralable in segment $[a,b]$, and assume that for every subsegment to $[a,b]$, exists $x$ so that $f(x)\le 1$.

Prove that $\int_a^bf(x)dx\le b-a$

I know that if for ALL points in the segment the assumption is true, but how do you go around "for every subsegment exists a point"?

Thank you!