Let $f : [0,\infty) \to \mathbb{R}$ be monotone-increasing. Then, it is well-known that the derivative $f'$ exists and takes nonnegative values almost everywhere on $[0,\infty)$.
Then, is it true that \begin{equation} f(b) - f(a) \geq \int_a^b f'(t) dt \text{ for "all" } 0 \leq a < b <\infty \end{equation} holds?
In Folland Real Analysis p. 108, exercise 33, he states the same result for $\mathbb{R}$ rather than $[0,\infty)$. I strongly suspect that the statement holds valid on $[0,\infty)$ without any change, but would like to make sure that I am correct.
Could anyone please clarify for me?