Does $\limsup_{x\to\infty}\int_x^{x+h}f'(y)dy = 0$, $\forall h>0$, imply $\lim_{x\to\infty}\int_x^{x+h}f'(y)dy = 0$, $\forall h>0$? I have an absolutely continuous function $f:[0,\infty)\to[0,\infty)$ that satisfies $\limsup_{x\to\infty}\int_x^{x+h}f'(y)dy = 0$ for all $h>0$. I need to check if it is true or false that $\lim_{x\to\infty}\int_x^{x+h}f'(y)dy = 0$ for all $h>0$. Any hint will be welcome.

... only if the second limit exists.  What's the infimum limit doing?  (This should be easy to determine.  Replace $f$ with $-f$ in your first argument.)  If the supremum and infimum limits agree, then the limit exists and agrees with them.
