Keilim, Chapter 13, Mishnah 1. A keli that breaks stops being a keli, and once it is no longer a keli it can no longer receive tumah. With metal kelim the test is functional: if the damage is severe enough that the object can no longer serve the purpose for which it was made, it is finished as a keli and it cannot become tameh. But if what remains still performs that original function, or still has a real connection to it, the remnant is a keli in its own right and is subject to tumah. Our Mishnah opens the chapter with a list of metal implements that broke, where the surviving pieces are still considered kelim.
The list itself runs as follows: a sword ("hasayif"); a knife ("vehasakin"); a dagger ("vehapigyon"), meaning a weapon whose single blade is ground sharp along both of its edges rather than along one alone (the term does not describe an implement with two blades protruding from two separate places); a spear ("veharomach"); a hand sickle ("magal yad"), the tool used to cut through wood and bone; a reaping sickle ("umagal katzir"), used on standing grain; a small clipper for hair ("vehashichor"); and the barbers' shears ("vehazug shel saparim"), which likewise cut hair but are larger than the small clipper. About every item on this list the Mishnah teaches: "shenechleku, harei eilu temei'in" - should any of them break into two, each of the two parts still stands as something that can contract tumah.
Why? Because every surviving portion can go on cutting, and cutting is exactly what the whole implement did before it broke. With a pair of shears, breaking means that the two blades have separated from each other, and each separated blade counts as a utensil on its own, for a lone blade of that sort is practically a knife. In the remaining items, the fracture ran across the width of the blade, leaving one segment attached to its handle and the other segment with nothing to hold it by. Because both pieces can still cut after a fashion, tumah applies to both.
Ribbi Yose disagrees, distinguishing between the two fragments according to which end of the blade they come from. The portion that is "hasamuch layad," the part left joined to the handle, he calls "tameh," since a person can genuinely go on cutting with it. But the fragment from the far end, "vehasamuch larosh," he calls "tahor," because realistically it will never be used: there is nothing to grip, and no one grasps a naked blade and exposes himself to being cut. With no practical use available to it, that piece has ceased to be a utensil.
It is worth being precise about where this argument applies. With broken shears there is no machloket at all. Everyone agrees that the two blades are now two kelim, since each blade comes with its own handle. The dispute concerns something like a knife or a sword that snapped into two pieces: the piece holding the handle is still serviceable, while according to Ribbi Yose the piece without a handle is not.
Next the Mishnah brings a dispute concerning a "misporet," a pincer-type implement that women used, roughly comparable to a tweezers. Each of its two arms, taken alone, is blunt; only when the arms are pressed together do they produce a cutting edge. The case discussed is "misporet shenechelkah lishnayim": the tool broke through the middle, leaving two separate arms that can no longer be closed against each other, so that all that remains is a pair of blunt pieces. On this the Mishnah records: "Ribbi Yehudah metamei, vachachamim metaherin." According to Ribbi Yehudah each arm is susceptible to tumah, since cutting with it is still possible. The chachamim maintain that each arm is tahor, for it has no sharp edge and would cut only with great effort. Having ceased to be a true utensil, it is outside the laws of tumah and stays tahor from that point onward.