@HeadShot
WARNING : i am not good at calculus, plz take others opinion also.
So as far as i know mod has no effect on continuity.
Because continuity means their is NO holes in the given function.
now consider a function which has a hole i.e.. its NOT continious, now by making the function positive ( |f(x)| ), the hole still exist, the only difference is that the hole now lies above X-axis instead of below.
so i think the property you read online is not true.