The cover of a householder's teni (a type of metal box which is susceptible to tumah),

made of metal (but is still tahor)

that one made into a mirror (by polishing it),

Rabbi Yehuda ruled is tahor (since it is still chiefly used as a lid and only occasionally as a mirror)

and the Sages ruled is tamei.

A [metal] mirror that broke

if it cannot reflect

the majority of a face

is tahor.

פרק יג משנה ז

Note that while a metal utensil that was broken is טהור מדאורייתא, it was decreed that if repaired it would regain its original level of טומאה, which is referred to as טומאה ישנה, or "old tumah"

Metal utensils (which, after becoming tamei from a corpse must be sprinkled with mei chattas, the water derived from a red heifer, on the third and seventh day after it became tamei)

contract tumah and become tahor when broken (that is to say that breaking a metal utensil that is tamei can have either the effect of accelerating and delaying its purification),

these are the words of Rabbi Eliezer (he held that if after the sprinkling of the third day it was broken and then remade, it may immediately be sprinkled again, instead of waiting for the seventh day; he was of the opinion that the decree of "old tumah" was intended only to prevent people from avoiding the process of using mei chattas, which this did).

Rabbi Yehoshua says

they do not become tahor except when whole (i.e. the regular process must be gone completed).

How is this?

If one sprinkled on them (on the third day after they became tamei),

they broke that day,

he remade them,

and returned and sprinkled on them a second time

that day,

those are tahor,

these are the words of Rabbi Eliezer;

Rabbi Yehoshua says

there is no [valid] sprinkling

before the third and seventh (i.e. it must be done as always, on the third and seventh day after it became tamei).