Why do characters in some art styles have a broken line for mouths? Kaname Madoka from Puella Magi Madoka Magica and Nakano Azusa from K-On! have a broken line for mouths.





Why do characters in some art styles have a broken line for mouths? The lines are broken unless it defines lips but I don't know otherwise.
I think the explanation given in moegamisama's answer is a step in the right direction; however, an argument purely from high-level principles about moe/cuteness lacks sufficient explanatory power to explain why this particular style ("broken line" mouths) is often seen. I think a more mechanistic explanation for why mouths are drawn this way is also necessary.

Taking for granted that the minimization of some facial features (a small nose, compensated by large eyes, say) is a common feature in "moe" character design today, and observing that the mouth is an oft-minimized feature, the mechanistic explanation, then, is as follows. 

Take a close look at the way Madoka's mouth is drawn in the picture you posted. You will notice that the thickness of the line that represents her mouth is not constant. Rather, it is thickest near the edges and becomes thinner as we approach the center of her face. The line "breaking" at the center of her face strikes me as an artistic convenience: rather than drawing a super-thin line there, the artist simply omits the line altogether, and the viewer's mind merely fills it in (this is the so-called "principle of closure" of gestalt psychology). You see the same thing (albeit less delicately) in the picture of Azusa.